From bcb6b4dcbcdafecfdeec7b5891882b0650708ee6 Mon Sep 17 00:00:00 2001
From: Mohamed Barakat <mohamed.barakat@rwth-aachen.de>
Date: Thu, 18 May 2023 14:35:39 +0200
Subject: [PATCH 1/4] use ObjectConstructor/MorphismConstructor

instead of FinSet/MapOfFinSets
---
 gap/SkeletalFinSets.gi | 74 +++++++++++++++++++++---------------------
 1 file changed, 37 insertions(+), 37 deletions(-)

diff --git a/gap/SkeletalFinSets.gi b/gap/SkeletalFinSets.gi
index 159bcd5..f5708fd 100644
--- a/gap/SkeletalFinSets.gi
+++ b/gap/SkeletalFinSets.gi
@@ -51,7 +51,7 @@ InstallMethod( CategoryOfSkeletalFinSets,
 end );
 
 ##
-InstallMethodForCompilerForCAP( FinSetOp,
+InstallMethod( FinSetOp,
         [ IsCategoryOfSkeletalFinSets, IsBigInt ],
         
   function ( cat, n )
@@ -96,7 +96,7 @@ InstallMethod( MapOfFinSets,
 end );
 
 ##
-InstallOtherMethodForCompilerForCAP( MapOfFinSets,
+InstallOtherMethod( MapOfFinSets,
         "for a category of skeletal finite sets, two CAP skeletal finite sets and a list",
         [ IsCategoryOfSkeletalFinSets, IsSkeletalFiniteSet, IsList, IsSkeletalFiniteSet ],
         
@@ -343,7 +343,7 @@ end );
 AddIdentityMorphism( SkeletalFinSets,
   function ( cat, n )
     
-    return MapOfFinSets( cat, n, [ 0 .. Length( n ) - 1 ], n );
+    return MorphismConstructor( cat, n, [ 0 .. Length( n ) - 1 ], n );
     
 end );
 
@@ -360,7 +360,7 @@ AddPreCompose( SkeletalFinSets,
     
     cmp := List( s, i -> im_post[1 + im_pre[1 + i]] );
     
-    return MapOfFinSets( cat, s, cmp, t );
+    return MorphismConstructor( cat, s, cmp, t );
     
 end );
 
@@ -368,7 +368,7 @@ end );
 AddImageObject( SkeletalFinSets,
   function ( cat, phi )
     
-    return FinSet( cat, BigInt( Length( Set( AsList( phi ) ) ) ) );
+    return ObjectConstructor( cat, BigInt( Length( Set( AsList( phi ) ) ) ) );
     
 end );
 
@@ -446,7 +446,7 @@ AddLift( SkeletalFinSets,
     gg := AsList( g );
     ff := AsList( f );
     
-    return MapOfFinSets( cat, S, List( S, x -> -1 + BigInt( SafePosition( gg, ff[1 + x] ) ) ), T );
+    return MorphismConstructor( cat, S, List( S, x -> -1 + BigInt( SafePosition( gg, ff[1 + x] ) ) ), T );
     
 end );
 
@@ -482,7 +482,7 @@ AddColift( SkeletalFinSets,
         return gg[SafePosition( ff, y )];
     end;
     
-    return MapOfFinSets( cat, S, List( S, y -> chi(y) ), T );
+    return MorphismConstructor( cat, S, List( S, y -> chi(y) ), T );
     
 end );
 
@@ -490,7 +490,7 @@ end );
 AddImageEmbeddingWithGivenImageObject( SkeletalFinSets,
   function ( cat, phi, image )
     
-    return MapOfFinSets( cat, image, Set( AsList( phi ) ), Range( phi ) );
+    return MorphismConstructor( cat, image, Set( AsList( phi ) ), Range( phi ) );
 
 end );
 
@@ -507,7 +507,7 @@ AddCoastrictionToImageWithGivenImageObject( SkeletalFinSets,
     
     L := List( s, i -> -1 + BigInt( SafePosition( images, G[1 + i] ) ) );
     
-    pi := MapOfFinSets( cat, s, L, image_object );
+    pi := MorphismConstructor( cat, s, L, image_object );
     
     #% CAP_JIT_DROP_NEXT_STATEMENT
     Assert( 3, IsEpimorphism( cat, pi ) );
@@ -531,7 +531,7 @@ end );
 AddTerminalObject( SkeletalFinSets,
   function ( cat )
     
-    return FinSet( cat, BigInt( 1 ) );
+    return ObjectConstructor( cat, BigInt( 1 ) );
     
 end );
 
@@ -539,7 +539,7 @@ end );
 AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject( SkeletalFinSets,
   function ( cat, m, t )
     
-    return MapOfFinSets( cat, m, ListWithIdenticalEntries( Length( m ), BigInt( 0 ) ), t );
+    return MorphismConstructor( cat, m, ListWithIdenticalEntries( Length( m ), BigInt( 0 ) ), t );
     
 end );
 
@@ -547,7 +547,7 @@ end );
 AddDirectProduct( SkeletalFinSets,
   function ( cat, L )
     
-    return FinSet( cat, Product( List( L, Length ) ) );
+    return ObjectConstructor( cat, Product( List( L, Length ) ) );
     
 end );
 
@@ -562,7 +562,7 @@ AddProjectionInFactorOfDirectProductWithGivenDirectProduct( SkeletalFinSets,
     
     a := Product( List( D{[ 1 .. k - 1 ]}, Length ) );
     
-    return MapOfFinSets( cat, P, List( P, i -> RemInt( QuoInt( i, a ), l ) ), T );
+    return MorphismConstructor( cat, P, List( P, i -> RemInt( QuoInt( i, a ), l ) ), T );
     
 end );
 
@@ -580,7 +580,7 @@ AddUniversalMorphismIntoDirectProductWithGivenDirectProduct( SkeletalFinSets,
     taus := List( tau, AsList );
     
     # if l = 0, then Sum( [ 0 .. l - 1 ], j -> ... ) = 0 ∈ TerminalObject = P
-    return MapOfFinSets( cat, T, List( T, i -> Sum( [ 0 .. l - 1 ], j -> taus[1 + j][1 + i] * dd[1 + j] ) ), P );
+    return MorphismConstructor( cat, T, List( T, i -> Sum( [ 0 .. l - 1 ], j -> taus[1 + j][1 + i] * dd[1 + j] ) ), P );
     
 end );
 
@@ -593,7 +593,7 @@ AddEqualizer( SkeletalFinSets,
     
     Eq := Filtered( [ 0 .. Length( s ) - 1 ], x -> ForAll( [ 1 .. Length( D ) - 1 ], j -> D2[j][1 + x] = D2[j + 1][1 + x] ) );
     
-    return FinSet( cat, Length( Eq ) );
+    return ObjectConstructor( cat, Length( Eq ) );
     
 end );
 
@@ -606,7 +606,7 @@ AddEmbeddingOfEqualizerWithGivenEqualizer( SkeletalFinSets,
     
     Eq := Filtered( [ 0 .. Length( s ) - 1 ], x -> ForAll( [ 1 .. Length( D ) - 1 ], j -> D2[j][1 + x] = D2[j + 1][1 + x] ) );
     
-    return MapOfFinSets( cat, E, Eq, s );
+    return MorphismConstructor( cat, E, Eq, s );
     
 end );
 
@@ -621,7 +621,7 @@ AddUniversalMorphismIntoEqualizerWithGivenEqualizer( SkeletalFinSets,
     
     t := AsList( tau );
     
-    return MapOfFinSets( cat, test_object, List( test_object, x -> -1 + BigInt( SafePosition( Eq, t[1 + x] ) ) ), E );
+    return MorphismConstructor( cat, test_object, List( test_object, x -> -1 + BigInt( SafePosition( Eq, t[1 + x] ) ) ), E );
     
 end );
 
@@ -640,7 +640,7 @@ end );
 AddInitialObject( SkeletalFinSets,
   function ( cat )
     
-    return FinSet( cat, BigInt( 0 ) );
+    return ObjectConstructor( cat, BigInt( 0 ) );
     
 end );
 
@@ -648,7 +648,7 @@ end );
 AddUniversalMorphismFromInitialObjectWithGivenInitialObject( SkeletalFinSets,
   function ( cat, m, I )
     
-    return MapOfFinSets( cat, I, [ ], m );
+    return MorphismConstructor( cat, I, [ ], m );
     
 end );
 
@@ -692,7 +692,7 @@ end );
 AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject( SkeletalFinSets,
   function ( cat, M, injective_object )
     
-    return MapOfFinSets( cat, M, [ 0 .. Length( M ) - 1 ], injective_object );
+    return MorphismConstructor( cat, M, [ 0 .. Length( M ) - 1 ], injective_object );
     
 end );
 
@@ -700,7 +700,7 @@ end );
 AddCoproduct( SkeletalFinSets,
   function ( cat, L )
     
-    return FinSet( cat, Sum( List( L, Length ) ) );
+    return ObjectConstructor( cat, Sum( List( L, Length ) ) );
     
 end );
 
@@ -715,7 +715,7 @@ AddInjectionOfCofactorOfCoproductWithGivenCoproduct( SkeletalFinSets,
     
     s := L[i];
     
-    return MapOfFinSets( cat, s, [ sum .. sum + Length( s ) - 1 ], coproduct );
+    return MorphismConstructor( cat, s, [ sum .. sum + Length( s ) - 1 ], coproduct );
     
 end );
 
@@ -726,7 +726,7 @@ AddUniversalMorphismFromCoproductWithGivenCoproduct( SkeletalFinSets,
     
     concat := Concatenation( List( tau, AsList ) );
     
-    return MapOfFinSets( cat, S, concat, test_object );
+    return MorphismConstructor( cat, S, concat, test_object );
     
 end );
 
@@ -734,7 +734,7 @@ end );
 AddCoequalizer( SkeletalFinSets,
   function ( cat, s, D )
   
-    return FinSet( cat, BigInt( Length( SKELETAL_FIN_SETS_ExplicitCoequalizer( s, D ) ) ) );
+    return ObjectConstructor( cat, BigInt( Length( SKELETAL_FIN_SETS_ExplicitCoequalizer( s, D ) ) ) );
     
 end );
 
@@ -747,7 +747,7 @@ AddProjectionOntoCoequalizerWithGivenCoequalizer( SkeletalFinSets,
     
     cmp := List( s, x -> -1 + BigInt( SafeUniquePositionProperty( Cq, c -> x in c ) ) );
     
-    return MapOfFinSets( cat, s, cmp, C );
+    return MorphismConstructor( cat, s, cmp, C );
     
 end );
 
@@ -758,7 +758,7 @@ AddUniversalMorphismFromCoequalizerWithGivenCoequalizer( SkeletalFinSets,
     
     Cq := SKELETAL_FIN_SETS_ExplicitCoequalizer( s, D );
 
-    return MapOfFinSets( cat, C, List( Cq, x -> tau( x[1] ) ), Range( tau ) );
+    return MorphismConstructor( cat, C, List( Cq, x -> tau( x[1] ) ), Range( tau ) );
     
 end );
 
@@ -805,7 +805,7 @@ AddCartesianBraidingInverseWithGivenDirectProducts( SkeletalFinSets,
     
     n := Length( N );
     
-    return MapOfFinSets( cat, MN, List( MN , i -> RemInt( i, n ) * m + QuoInt( i, n ) ), NM );
+    return MorphismConstructor( cat, MN, List( MN , i -> RemInt( i, n ) * m + QuoInt( i, n ) ), NM );
     
 end );
 
@@ -817,7 +817,7 @@ AddExponentialOnObjects( SkeletalFinSets,
     m := Length( M );
     n := Length( N );
     
-    return FinSet( cat, n ^ m );
+    return ObjectConstructor( cat, n ^ m );
     
 end );
 
@@ -831,7 +831,7 @@ AddCartesianLambdaElimination( SkeletalFinSets,
     
     v := AsList( intro )[1];
     
-    return MapOfFinSets( cat,
+    return MorphismConstructor( cat,
                    M,
                    List( [ 0 .. m - 1 ], i -> RemInt( QuoInt( v, n^i ), n ) ),
                    N );
@@ -850,7 +850,7 @@ AddCartesianLambdaIntroduction( SkeletalFinSets,
     
     images := AsList( map );
     
-    return MapOfFinSets( cat,
+    return MorphismConstructor( cat,
                    TerminalObject( cat ),
                    [ Sum( List( [ 0 .. m - 1 ], k -> images[1 + k] * n^k ) ) ],
                    ExponentialOnObjects( cat, M, N ) );
@@ -864,7 +864,7 @@ AddExactCoverWithGlobalElements( SkeletalFinSets,
     
     T := TerminalObject( cat );
     
-    return List( [ 0 .. Length( A ) - 1 ], i -> MapOfFinSets( cat, T, [ i ], A ) );
+    return List( [ 0 .. Length( A ) - 1 ], i -> MorphismConstructor( cat, T, [ i ], A ) );
     
 end );
 
@@ -880,7 +880,7 @@ AddExponentialOnMorphismsWithGivenExponentials( SkeletalFinSets,
 
     mors := ExactCoverWithGlobalElements( cat, MN );
     
-    return MapOfFinSets( cat,
+    return MorphismConstructor( cat,
                    S,
                    List( mors, mor ->
                          AsList( CartesianLambdaIntroduction( cat,
@@ -911,7 +911,7 @@ AddCartesianEvaluationMorphismWithGivenSource( SkeletalFinSets,
     
     exp := n ^ m;
     
-    return MapOfFinSets( cat, HM_NxM, List( [ 0 .. ( n^m * m ) - 1 ], i -> RemInt( QuoInt( i, n^QuoInt( i, exp ) ), n ) ), N );
+    return MorphismConstructor( cat, HM_NxM, List( [ 0 .. ( n^m * m ) - 1 ], i -> RemInt( QuoInt( i, n^QuoInt( i, exp ) ), n ) ), N );
     
 end );
 
@@ -925,9 +925,9 @@ AddCartesianCoevaluationMorphismWithGivenRange( SkeletalFinSets,
     
     mn := m * n;
     
-    #return MapOfFinSets( cat, M, List( [ 0 .. m - 1 ], i -> Sum( [ 0 .. n - 1 ], j -> ( i + m * j ) * (m*n)^j ) ), HN_MxN );
+    #return MorphismConstructor( cat, M, List( [ 0 .. m - 1 ], i -> Sum( [ 0 .. n - 1 ], j -> ( i + m * j ) * (m*n)^j ) ), HN_MxN );
     
-    return MapOfFinSets( cat, M, List( [ 0 .. m - 1 ], i -> i * GeometricSum( mn, n ) + m * mn * GeometricSumDiff1( mn, n ) ), HN_MxN );
+    return MorphismConstructor( cat, M, List( [ 0 .. m - 1 ], i -> i * GeometricSum( mn, n ) + m * mn * GeometricSumDiff1( mn, n ) ), HN_MxN );
     
 end );
 
@@ -935,7 +935,7 @@ end );
 AddSubobjectClassifier( SkeletalFinSets,
   function ( cat )
       
-      return FinSet( cat, BigInt( 2 ) );
+      return ObjectConstructor( cat, BigInt( 2 ) );
       
 end );
 
@@ -959,7 +959,7 @@ AddClassifyingMorphismOfSubobjectWithGivenSubobjectClassifier( SkeletalFinSets,
                    
                end );
       
-      return MapOfFinSets( cat, range, chi, Omega );
+      return MorphismConstructor( cat, range, chi, Omega );
       
 end );
 

From a0decc0e5e1e392aaa2b47a0896048bc1dd03a08 Mon Sep 17 00:00:00 2001
From: Mohamed Barakat <mohamed.barakat@rwth-aachen.de>
Date: Fri, 19 May 2023 23:05:44 +0200
Subject: [PATCH 2/4] converted logic templates into logic functions

Co-authored-by: Fabian Zickgraf <fabian.zickgraf@uni-siegen.de>
---
 gap/CompilerLogic.gi | 90 ++++++++++++++++++++++++++++++++------------
 1 file changed, 66 insertions(+), 24 deletions(-)

diff --git a/gap/CompilerLogic.gi b/gap/CompilerLogic.gi
index 3de50c7..955fb3b 100644
--- a/gap/CompilerLogic.gi
+++ b/gap/CompilerLogic.gi
@@ -85,6 +85,72 @@ CapJitAddLogicFunction( function ( tree )
     
 end );
 
+## list{[ ]} => [ ]
+CapJitAddLogicFunction( function ( tree )
+  local pre_func;
+    
+    Info( InfoCapJit, 1, "####" );
+    Info( InfoCapJit, 1, "Apply logic for extracting the empty sublist of a given list" );
+    
+    pre_func :=
+      function ( tree, additional_arguments )
+        local args;
+        
+        if CapJitIsCallToGlobalFunction( tree, "{}" ) then
+            
+            args := tree.args;
+            
+            if args.2.type = "EXPR_LIST" and args.2.list.length = 0 and IsBound( args.1.data_type ) then
+                
+                return rec( type := "EXPR_LIST", list := AsSyntaxTreeList( [ ] ), data_type := args.1.data_type );
+                
+            fi;
+            
+        fi;
+        
+        return tree;
+        
+    end;
+    
+    return CapJitIterateOverTree( tree, pre_func, CapJitResultFuncCombineChildren, ReturnTrue, true );
+    
+end );
+
+## Product( [ ] ) => BigInt( 1 ) and Product( [ ], func ) => BigInt( 1 )
+CapJitAddLogicFunction( function ( tree )
+  local pre_func;
+    
+    Info( InfoCapJit, 1, "####" );
+    Info( InfoCapJit, 1, "Apply logic for computing the empty product of an empty list of integers" );
+    
+    pre_func :=
+      function ( tree, additional_arguments )
+        local args, is_big_int, values;
+        
+        if CapJitIsCallToGlobalFunction( tree, "Product" ) then
+            
+            args := tree.args;
+            
+            if args.1.type = "EXPR_LIST" and args.1.list.length = 0 and IsBound( args.1.data_type ) then
+                
+                return rec( type := "EXPR_FUNCCALL",
+                            funcref := rec(
+                                    gvar := "BigInt",
+                                    type := "EXPR_REF_GVAR" ),
+                            args := rec( 1 := rec( type := "EXPR_INT", value := 1 ), length := 1, type := "SYNTAX_TREE_LIST" ) );
+                
+            fi;
+            
+        fi;
+        
+        return tree;
+        
+    end;
+    
+    return CapJitIterateOverTree( tree, pre_func, CapJitResultFuncCombineChildren, ReturnTrue, true );
+    
+end );
+
 ## Teach CompilerForCAP about the input type of the function so it can correctly type the function,
 ## that can be done with the following code (adapted from the existing List type signature):
 ##
@@ -260,14 +326,6 @@ CapJitAddLogicTemplate(
     )
 );
 
-CapJitAddLogicTemplate(
-    rec(
-        variable_names := [ "list" ],
-        src_template := "list{[ ]}",
-        dst_template := "[ ]",
-    )
-);
-
 CapJitAddLogicTemplate(
     rec(
         variable_names := [ "list" ],
@@ -284,22 +342,6 @@ CapJitAddLogicTemplate(
     )
 );
 
-CapJitAddLogicTemplate(
-    rec(
-        variable_names := [ ],
-        src_template := "Product( [ ] )",
-        dst_template := "BigInt( 1 )",
-    )
-);
-
-CapJitAddLogicTemplate(
-    rec(
-        variable_names := [ "func" ],
-        src_template := "Product( [ ], func )",
-        dst_template := "BigInt( 1 )",
-    )
-);
-
 CapJitAddLogicTemplate(
     rec(
         variable_names := [ "entry" ],

From 0fafabb97c4f2b30a70bcb9953c75658dc3133ca Mon Sep 17 00:00:00 2001
From: Mohamed Barakat <mohamed.barakat@rwth-aachen.de>
Date: Sat, 20 May 2023 16:58:47 +0200
Subject: [PATCH 3/4] renamed IsSkeletalFiniteSet(Map) ->
 IsObject/MorphismInCategoryOfSkeletalFinSets

---
 gap/CompilerLogic.gi   |  4 ++--
 gap/Julia.gi           |  2 +-
 gap/SkeletalFinSets.gd | 32 ++++++++++++++++----------------
 gap/SkeletalFinSets.gi | 34 +++++++++++++++++-----------------
 4 files changed, 36 insertions(+), 36 deletions(-)

diff --git a/gap/CompilerLogic.gi b/gap/CompilerLogic.gi
index 955fb3b..8102680 100644
--- a/gap/CompilerLogic.gi
+++ b/gap/CompilerLogic.gi
@@ -154,7 +154,7 @@ end );
 ## Teach CompilerForCAP about the input type of the function so it can correctly type the function,
 ## that can be done with the following code (adapted from the existing List type signature):
 ##
-CapJitAddTypeSignature( "List", [ IsSkeletalFiniteSet, IsFunction ], function ( args, func_stack )
+CapJitAddTypeSignature( "List", [ IsObjectInCategoryOfSkeletalFinSets, IsFunction ], function ( args, func_stack )
     
     args := ShallowCopy( args );
     
@@ -175,7 +175,7 @@ end );
 CapJitAddLogicTemplate(
     rec(
         variable_names := [ "M", "func" ],
-        variable_filters := [ IsSkeletalFiniteSet, IsObject ],
+        variable_filters := [ IsObjectInCategoryOfSkeletalFinSets, IsObject ],
         src_template := "List( M, func )",
         dst_template := "List( [ 0 .. Length( M ) - 1 ], func )",
     )
diff --git a/gap/Julia.gi b/gap/Julia.gi
index ef26785..8b1e543 100644
--- a/gap/Julia.gi
+++ b/gap/Julia.gi
@@ -33,7 +33,7 @@ end );
 
 ##
 InstallMethod( MapOfFinSets,
-        [ IsSkeletalFiniteSet, IsJuliaObject, IsSkeletalFiniteSet ],
+        [ IsObjectInCategoryOfSkeletalFinSets, IsJuliaObject, IsObjectInCategoryOfSkeletalFinSets ],
         
   function ( source, graph, range )
     
diff --git a/gap/SkeletalFinSets.gd b/gap/SkeletalFinSets.gd
index 17fe281..6279abe 100644
--- a/gap/SkeletalFinSets.gd
+++ b/gap/SkeletalFinSets.gd
@@ -19,14 +19,14 @@ DeclareCategory( "IsCategoryOfSkeletalFinSets",
 #! The GAP category of objects in the category
 #! of skeletal finite sets.
 #! @Arguments object
-DeclareCategory( "IsSkeletalFiniteSet",
+DeclareCategory( "IsObjectInCategoryOfSkeletalFinSets",
                  IsCapCategoryObject );
 
 #! @Description
 #! The GAP category of morphisms in the category
 #! of skeletal finite sets.
 #! @Arguments object
-DeclareCategory( "IsSkeletalFiniteSetMap",
+DeclareCategory( "IsMorphismInCategoryOfSkeletalFinSets",
                  IsCapCategoryMorphism );
 
 #! @Section Attributes
@@ -37,9 +37,9 @@ DeclareCategory( "IsSkeletalFiniteSetMap",
 #! @Arguments M
 #! @Returns an integer
 DeclareAttribute( "Length",
-        IsSkeletalFiniteSet );
+        IsObjectInCategoryOfSkeletalFinSets );
 
-CapJitAddTypeSignature( "Length", [ IsSkeletalFiniteSet ], IsBigInt );
+CapJitAddTypeSignature( "Length", [ IsObjectInCategoryOfSkeletalFinSets ], IsBigInt );
 
 #! @Description
 #!  The list associated to a skeletal finite set, i.e.,
@@ -47,18 +47,18 @@ CapJitAddTypeSignature( "Length", [ IsSkeletalFiniteSet ], IsBigInt );
 #! @Arguments M
 #! @Returns a list
 DeclareAttribute( "AsList",
-        IsSkeletalFiniteSet );
+        IsObjectInCategoryOfSkeletalFinSets );
 
-CapJitAddTypeSignature( "AsList", [ IsSkeletalFiniteSet ], rec( filter := IsList, element_type := rec( filter := IsBigInt ) ) );
+CapJitAddTypeSignature( "AsList", [ IsObjectInCategoryOfSkeletalFinSets ], rec( filter := IsList, element_type := rec( filter := IsBigInt ) ) );
 
 #! @Description
-#!  The graph defining the skeletal finite set morphism <A>phi</A>, see <Ref Oper="MapOfFinSets" Label="for IsSkeletalFiniteSet, IsList, IsSkeletalFiniteSet" />.
+#!  The graph defining the skeletal finite set morphism <A>phi</A>, see <Ref Oper="MapOfFinSets" Label="for IsObjectInCategoryOfSkeletalFinSets, IsList, IsObjectInCategoryOfSkeletalFinSets" />.
 #! @Arguments phi
 #! @Returns a list
 DeclareAttribute( "AsList",
-        IsSkeletalFiniteSetMap );
+        IsMorphismInCategoryOfSkeletalFinSets );
 
-CapJitAddTypeSignature( "AsList", [ IsSkeletalFiniteSetMap ], rec( filter := IsList, element_type := rec( filter := IsBigInt ) ) );
+CapJitAddTypeSignature( "AsList", [ IsMorphismInCategoryOfSkeletalFinSets ], rec( filter := IsList, element_type := rec( filter := IsBigInt ) ) );
 
 #! @Section Constructors
 
@@ -97,7 +97,7 @@ KeyDependentOperation( "FinSet", IsCategoryOfSkeletalFinSets, IsBigInt, ReturnTr
 #! @Arguments s, G, t
 #! @Returns a &CAP; morphism
 DeclareOperation( "MapOfFinSets",
-        [ IsSkeletalFiniteSet, IsList, IsSkeletalFiniteSet ] );
+        [ IsObjectInCategoryOfSkeletalFinSets, IsList, IsObjectInCategoryOfSkeletalFinSets ] );
 #! @InsertChunk  SkeletalMapOfFinSets
 
 #! @Section Tools
@@ -107,7 +107,7 @@ DeclareOperation( "MapOfFinSets",
 #! @Arguments s, f
 #! @Returns a list
 DeclareOperation( "ListOp",
-        [ IsSkeletalFiniteSet, IsFunction ] );
+        [ IsObjectInCategoryOfSkeletalFinSets, IsFunction ] );
 
 #! @Description
 #!  Construct the embedding $\iota:$<A>s</A>$\to$<A>t</A> of the finite sets <A>s</A> and <A>t</A>,
@@ -115,34 +115,34 @@ DeclareOperation( "ListOp",
 #! @Arguments s, t
 #! @Returns a &CAP; morphism
 DeclareOperation( "EmbeddingOfFinSets",
-        [ IsSkeletalFiniteSet, IsSkeletalFiniteSet ] );
+        [ IsObjectInCategoryOfSkeletalFinSets, IsObjectInCategoryOfSkeletalFinSets ] );
 
 #! @Description
 #!  Compute the Preimage of <A>t</A> under the morphism <A>phi</A>.
 #! @Arguments phi, t
 #! @Returns a &CAP; object
 DeclareOperation( "Preimage",
-        [ IsSkeletalFiniteSetMap, IsList ] );
+        [ IsMorphismInCategoryOfSkeletalFinSets, IsList ] );
 
 #! @Description
 #!  Compute the image of <A>s_</A> under the morphism <A>phi</A>.
 #! @Arguments phi, s_
 #! @Returns a &CAP; object
 DeclareOperation( "ImageObject",
-        [ IsSkeletalFiniteSetMap, IsSkeletalFiniteSet ] );
+        [ IsMorphismInCategoryOfSkeletalFinSets, IsObjectInCategoryOfSkeletalFinSets ] );
 
 #! @Description
 #!  Returns the image of <C><A>L</A>[1]</C> under the map <A>phi</A> assuming <C><A>L</A>[1]</C> is a nonnegative integer smaller than <C>Length( Source( <A>phi</A> ) )</C>.
 #! @Arguments phi, L
 #! @Returns a list
 # DeclareOperation( "CallFuncList",
-#         [ IsSkeletalFiniteSetMap, IsList ] );
+#         [ IsMorphismInCategoryOfSkeletalFinSets, IsList ] );
 
 # Technical functions
 DeclareGlobalFunction( "INSTALL_FUNCTIONS_FOR_SKELETAL_FIN_SETS" );
 
 DeclareGlobalFunction( "SKELETAL_FIN_SETS_ExplicitCoequalizer" );
-CapJitAddTypeSignature( "SKELETAL_FIN_SETS_ExplicitCoequalizer", [ IsSkeletalFiniteSet, IsList ], rec( filter := IsList, element_type := rec( filter := IsList, element_type := rec( filter := IsBigInt ) ) ) );
+CapJitAddTypeSignature( "SKELETAL_FIN_SETS_ExplicitCoequalizer", [ IsObjectInCategoryOfSkeletalFinSets, IsList ], rec( filter := IsList, element_type := rec( filter := IsList, element_type := rec( filter := IsBigInt ) ) ) );
 
 DeclareGlobalFunction( "SKELETAL_FIN_SETS_IsMonomorphism" );
 CapJitAddTypeSignature( "SKELETAL_FIN_SETS_IsMonomorphism", [ IsList, IsBigInt ], IsBool );
diff --git a/gap/SkeletalFinSets.gi b/gap/SkeletalFinSets.gi
index f5708fd..87331a6 100644
--- a/gap/SkeletalFinSets.gi
+++ b/gap/SkeletalFinSets.gi
@@ -13,7 +13,7 @@ InstallMethod( CategoryOfSkeletalFinSets,
     
     cat := CreateCapCategoryWithDataTypes(
         "SkeletalFinSets", IsCategoryOfSkeletalFinSets,
-        IsSkeletalFiniteSet, IsSkeletalFiniteSetMap, IsCapCategoryTwoCell,
+        IsObjectInCategoryOfSkeletalFinSets, IsMorphismInCategoryOfSkeletalFinSets, IsCapCategoryTwoCell,
         IsBigInt, rec( filter := IsList, element_type := rec( filter := IsBigInt ) ), fail
     );
     
@@ -63,7 +63,7 @@ end );
 ##
 InstallMethod( AsList,
         "for a CAP skeletal finite set",
-        [ IsSkeletalFiniteSet ],
+        [ IsObjectInCategoryOfSkeletalFinSets ],
         
   function ( s )
     
@@ -74,7 +74,7 @@ end );
 ##
 InstallMethod( ListOp,
         "for a CAP skeletal finite set and a function",
-        [ IsSkeletalFiniteSet, IsFunction ],
+        [ IsObjectInCategoryOfSkeletalFinSets, IsFunction ],
         
   function ( s, f )
     
@@ -87,7 +87,7 @@ end );
 ##
 InstallMethod( MapOfFinSets,
         "for two CAP skeletal finite sets and a list",
-        [ IsSkeletalFiniteSet, IsList, IsSkeletalFiniteSet ],
+        [ IsObjectInCategoryOfSkeletalFinSets, IsList, IsObjectInCategoryOfSkeletalFinSets ],
         
   function ( s, G, t )
     
@@ -98,7 +98,7 @@ end );
 ##
 InstallOtherMethod( MapOfFinSets,
         "for a category of skeletal finite sets, two CAP skeletal finite sets and a list",
-        [ IsCategoryOfSkeletalFinSets, IsSkeletalFiniteSet, IsList, IsSkeletalFiniteSet ],
+        [ IsCategoryOfSkeletalFinSets, IsObjectInCategoryOfSkeletalFinSets, IsList, IsObjectInCategoryOfSkeletalFinSets ],
         
   function ( cat, s, G, t )
     
@@ -109,7 +109,7 @@ end );
 ##
 InstallMethod( EmbeddingOfFinSets,
         "for two CAP skeletal finite sets",
-        [ IsSkeletalFiniteSet, IsSkeletalFiniteSet ],
+        [ IsObjectInCategoryOfSkeletalFinSets, IsObjectInCategoryOfSkeletalFinSets ],
         
   function ( s, t )
     local iota;
@@ -126,7 +126,7 @@ end );
 ##
 InstallMethod( Preimage,
         "for a CAP map of skeletal finite sets and a CAP skeletal finite set",
-        [ IsSkeletalFiniteSetMap, IsList ],
+        [ IsMorphismInCategoryOfSkeletalFinSets, IsList ],
         
   function ( phi, t )
     local S;
@@ -142,7 +142,7 @@ end );
 ##
 InstallMethod( ImageObject,
      "for a CAP map of skeletal finite sets and a CAP skeletal finite set",
-     [ IsSkeletalFiniteSetMap, IsSkeletalFiniteSet ],
+     [ IsMorphismInCategoryOfSkeletalFinSets, IsObjectInCategoryOfSkeletalFinSets ],
       function ( phi, s_ )
 
     return ImageObject( PreCompose( EmbeddingOfFinSets( s_, Source( phi ) ), phi ) );
@@ -152,7 +152,7 @@ end );
 ##
 InstallMethod( CallFuncList,
         "for a CAP map of skeletal finite sets and a list",
-    [ IsSkeletalFiniteSetMap, IsList ],
+    [ IsMorphismInCategoryOfSkeletalFinSets, IsList ],
         
   function ( phi, L )
     local x;
@@ -988,7 +988,7 @@ end );
 ##
 InstallMethod( String,
         "for a CAP skeletal finite set",
-        [ IsSkeletalFiniteSet ],
+        [ IsObjectInCategoryOfSkeletalFinSets ],
         
   function ( s )
     return Concatenation( "FinSet( SkeletalFinSets, ", String( Length( s ) ), " )" );
@@ -997,7 +997,7 @@ end );
 ##
 InstallMethod( String,
         "for a CAP map of skeletal finite sets",
-        [ IsSkeletalFiniteSetMap ],
+        [ IsMorphismInCategoryOfSkeletalFinSets ],
         
   function ( phi )
     return Concatenation( "MapOfFinSets( SkeletalFinSets, ", String( Source( phi ) ), ", ", String( AsList( phi ) ), ", ", String( Range( phi ) ), " )" );
@@ -1006,7 +1006,7 @@ end );
 ##
 InstallMethod( ViewString,
         "for a CAP skeletal finite set",
-        [ IsSkeletalFiniteSet ],
+        [ IsObjectInCategoryOfSkeletalFinSets ],
         
   function ( s )
     return Concatenation( "|", String( Length( s ) ), "|" );
@@ -1015,7 +1015,7 @@ end );
 ##
 InstallMethod( ViewString,
         "for a CAP map of skeletal finite sets",
-        [ IsSkeletalFiniteSetMap ],
+        [ IsMorphismInCategoryOfSkeletalFinSets ],
         
   function ( phi )
     local arrow;
@@ -1049,7 +1049,7 @@ end );
 ##
 InstallMethod( PrintString,
         "for a CAP skeletal finite set",
-        [ IsSkeletalFiniteSet ],
+        [ IsObjectInCategoryOfSkeletalFinSets ],
         
   function ( s )
     local l, string;
@@ -1073,7 +1073,7 @@ end );
 ##
 InstallMethod( PrintString,
         "for a CAP map of skeletal finite sets",
-        [ IsSkeletalFiniteSetMap ],
+        [ IsMorphismInCategoryOfSkeletalFinSets ],
         
   function ( phi )
     
@@ -1087,7 +1087,7 @@ end );
 ##
 InstallMethod( DisplayString,
         "for a CAP skeletal finite set",
-        [ IsSkeletalFiniteSet ],
+        [ IsObjectInCategoryOfSkeletalFinSets ],
         
   function ( s )
     
@@ -1098,7 +1098,7 @@ end );
 ##
 InstallMethod( DisplayString,
         "for a CAP map of skeletal finite sets",
-        [ IsSkeletalFiniteSetMap ],
+        [ IsMorphismInCategoryOfSkeletalFinSets ],
         
   function ( phi )
     

From 30c9d082f299b24a5076919601b1655c33b81e8f Mon Sep 17 00:00:00 2001
From: Mohamed Barakat <mohamed.barakat@rwth-aachen.de>
Date: Sun, 14 May 2023 20:14:46 +0200
Subject: [PATCH 4/4] functional

---
 ...etalFinSetsWithMorphismsGivenByFunctions.g |   55 +
 ...SkeletalFinSetsWithMorphismsGivenByLists.g |    7 +-
 examples/SkeletalWellDefined.g                |    4 -
 gap/CompilerLogic.gi                          |  124 ++
 gap/SkeletalFinSets.gd                        |   58 +-
 gap/SkeletalFinSets.gi                        |  466 +++++---
 gap/SkeletalFinSetsEager.gd                   |   64 ++
 gap/SkeletalFinSetsEager.gi                   |  282 +++++
 gap/init.gi                                   |    8 +-
 ...ithMorphismsGivenByFunctionsPrecompiled.gi | 1016 +++++++++++++++++
 ...etsWithMorphismsGivenByListsPrecompiled.gi |  287 +++--
 init.g                                        |    2 +
 read.g                                        |    4 +
 13 files changed, 2088 insertions(+), 289 deletions(-)
 create mode 100644 examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions.g
 create mode 100644 gap/SkeletalFinSetsEager.gd
 create mode 100644 gap/SkeletalFinSetsEager.gi
 create mode 100644 gap/precompiled_categories/CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled.gi

diff --git a/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions.g b/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions.g
new file mode 100644
index 0000000..bedded4
--- /dev/null
+++ b/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions.g
@@ -0,0 +1,55 @@
+#! @Chapter Precompilation
+
+#! @Section Precompiling the category of skeletal finite sets
+
+#! @Example
+
+#! #@if ValueOption( "no_precompiled_code" ) <> true
+
+LoadPackage( "FinSetsForCAP", false );
+#! true
+LoadPackage( "CompilerForCAP", false );
+#! true
+
+ReadPackageOnce( "FinSetsForCAP", "gap/CompilerLogic.gi" );
+#! true
+
+category_constructor :=
+  {} -> CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions( );;
+given_arguments := [ ];;
+compiled_category_name :=
+  "CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled";;
+package_name := "FinSetsForCAP";;
+primitive_operations :=
+  ListPrimitivelyInstalledOperationsOfCategory(
+          category_constructor( : no_precompiled_code := true ) );;
+list_of_operations :=
+  SortedList( Concatenation( primitive_operations, [
+          #"HasPushoutComplement",
+          "PushoutComplement",
+          ] ) );;
+
+CapJitPrecompileCategoryAndCompareResult(
+        category_constructor,
+        given_arguments,
+        package_name,
+        compiled_category_name
+        : operations := list_of_operations,
+        number_of_objectified_objects_in_data_structure_of_object := 1,
+        number_of_objectified_morphisms_in_data_structure_of_object := 0,
+        number_of_objectified_objects_in_data_structure_of_morphism := 2,
+        number_of_objectified_morphisms_in_data_structure_of_morphism := 1
+);;
+
+CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled( );
+#! SkeletalFinSets
+
+cat := CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions( );
+#! SkeletalFinSets
+
+cat!.precompiled_functions_added;
+#! true
+
+#! #@fi
+
+#! @EndExample
diff --git a/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByLists.g b/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByLists.g
index 900c6e3..5e2ae6a 100644
--- a/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByLists.g
+++ b/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByLists.g
@@ -11,10 +11,11 @@ LoadPackage( "FinSetsForCAP", false );
 LoadPackage( "CompilerForCAP", false );
 #! true
 
-ReadPackage( "FinSetsForCAP", "gap/CompilerLogic.gi" );
+ReadPackageOnce( "FinSetsForCAP", "gap/CompilerLogic.gi" );
 #! true
 
-category_constructor := {} -> CategoryOfSkeletalFinSets( );;
+category_constructor :=
+  {} -> CategoryOfSkeletalFinSetsWithMorphismsGivenByLists( );;
 given_arguments := [ ];;
 compiled_category_name :=
   "CategoryOfSkeletalFinSetsWithMorphismsGivenByListsPrecompiled";;
@@ -43,7 +44,7 @@ CapJitPrecompileCategoryAndCompareResult(
 CategoryOfSkeletalFinSetsWithMorphismsGivenByListsPrecompiled( );
 #! SkeletalFinSets
 
-cat := CategoryOfSkeletalFinSets( );
+cat := CategoryOfSkeletalFinSetsWithMorphismsGivenByLists( );
 #! SkeletalFinSets
 
 cat!.precompiled_functions_added;
diff --git a/examples/SkeletalWellDefined.g b/examples/SkeletalWellDefined.g
index 506574a..6f53fa8 100644
--- a/examples/SkeletalWellDefined.g
+++ b/examples/SkeletalWellDefined.g
@@ -7,10 +7,6 @@ s := FinSet( 7 );
 #! |7|
 t := FinSet( 4 );
 #! |4|
-psi := MapOfFinSets( s, [ 0, 2, 1, 2, 1, 3 ], t );
-#! |7| → |4|
-IsWellDefined( psi );
-#! false
 psi := MapOfFinSets( s, [ 0, 2, 1, 2, 1, 3, -2 ], t );
 #! |7| → |4|
 IsWellDefined( psi );
diff --git a/gap/CompilerLogic.gi b/gap/CompilerLogic.gi
index 8102680..c23dc45 100644
--- a/gap/CompilerLogic.gi
+++ b/gap/CompilerLogic.gi
@@ -116,6 +116,51 @@ CapJitAddLogicFunction( function ( tree )
     
 end );
 
+## [ entry1, ... ]{[ ]} => [ ]
+CapJitAddLogicFunction( function ( tree )
+  local pre_func;
+    
+    Info( InfoCapJit, 1, "####" );
+    Info( InfoCapJit, 1, "Apply logic for extracting the empty sublist of a given list" );
+    
+    pre_func :=
+      function ( tree, additional_arguments )
+        local args, is_big_int, values;
+        
+        if CapJitIsCallToGlobalFunction( tree, "{}" ) then
+            
+            args := tree.args;
+            
+            if args.2.type = "EXPR_LIST" and args.2.list.length = 0 then
+                
+                if IsBound( args.1.list ) and IsBound( args.1.list.1 ) then
+                    
+                    if IsBound( args.1.list.1.data_type ) then
+                        return rec( type := "EXPR_LIST", list := AsSyntaxTreeList( [ ] ), data_type := args.1.list.1.data_type );
+                    elif IsBound( args.1.list.1.funcref ) and IsBound( args.1.list.1.funcref.data_type ) then
+                        return rec( type := "EXPR_LIST", list := AsSyntaxTreeList( [ ] ), data_type := args.1.list.1.funcref.data_type.signature[2] );
+                    elif IsBound( args.1.list.1.funcref ) and CapJitIsCallToGlobalFunction( args.1.list.1, "Length" ) then
+                        return rec( type := "EXPR_LIST", list := AsSyntaxTreeList( [ ] ), data_type := rec( filter := IsInt ) );
+                    fi;
+                    
+                elif CapJitIsCallToGlobalFunction( args.1, "ListWithIdenticalEntries" ) then
+                    
+                    return rec( type := "EXPR_LIST", list := AsSyntaxTreeList( [ ] ), data_type := args.1.args.2.funcref.data_type.signature[2] );
+                    
+                fi;
+                
+            fi;
+            
+        fi;
+        
+        return tree;
+        
+    end;
+    
+    return CapJitIterateOverTree( tree, pre_func, CapJitResultFuncCombineChildren, ReturnTrue, true );
+    
+end );
+
 ## Product( [ ] ) => BigInt( 1 ) and Product( [ ], func ) => BigInt( 1 )
 CapJitAddLogicFunction( function ( tree )
   local pre_func;
@@ -171,6 +216,15 @@ CapJitAddTypeSignature( "List", [ IsObjectInCategoryOfSkeletalFinSets, IsFunctio
     
 end );
 
+##
+CapJitAddLogicTemplate(
+    rec(
+        variable_names := [ "list" ],
+        src_template := "List( list, ID_FUNC )",
+        dst_template := "list",
+    )
+);
+
 ##
 CapJitAddLogicTemplate(
     rec(
@@ -326,6 +380,15 @@ CapJitAddLogicTemplate(
     )
 );
 
+CapJitAddLogicTemplate(
+    rec(
+        variable_names := [ "number1", "number2" ],
+        variable_filters := [ IsBigInt, IsBigInt ],
+        src_template := "REM_INT( REM_INT( number1, number2 ), number2 )",
+        dst_template := "REM_INT( number1, number2 )",
+    )
+);
+
 CapJitAddLogicTemplate(
     rec(
         variable_names := [ "list" ],
@@ -382,6 +445,14 @@ CapJitAddLogicTemplate(
     )
 );
 
+CapJitAddLogicTemplate(
+    rec(
+        variable_names := [ "entry" ],
+        src_template := "ListWithIdenticalEntries( BigInt( 1 ), entry )",
+        dst_template := "[ entry ]",
+    )
+);
+
 CapJitAddLogicTemplate(
     rec(
         variable_names := [ "length", "constant", "pos" ],
@@ -467,3 +538,56 @@ CapJitAddLogicTemplate(
         dst_template := "entry in list",
     )
 );
+
+CapJitAddLogicTemplate(
+    rec(
+        variable_names := [ "map" ],
+        variable_filters := [ IsMorphismInCategoryOfSkeletalFinSets ],
+        src_template := "List( [ 0 .. Length( Source( map ) ) - 1 ], i -> AsList( map )[1 + i] )",
+        dst_template := "AsList( map )",
+    )
+);
+
+CapJitAddLogicTemplate(
+    rec(
+        variable_names := [ "map" ],
+        variable_filters := [ IsMorphismInCategoryOfSkeletalFinSets ],
+        src_template := "AsList( map ){[ 1 .. Length( Source( map ) ) ]}",
+        dst_template := "AsList( map )",
+    )
+);
+
+CapJitAddLogicTemplate(
+    rec(
+        variable_names := [ "list", "number" ],
+        variable_filters := [ IsList, IsBigInt ],
+        src_template := "List( [ 0 .. number - 1 ], i -> list[1 + i] )",
+        dst_template := "list{[ 1 .. number ]}",
+    )
+);
+
+CapJitAddLogicTemplate(
+    rec(
+        variable_names := [ "length", "offset" ],
+        variable_filters := [ IsBigInt, IsBigInt ],
+        src_template := "List( [ 0 .. length - 1 ], i -> offset + i )",
+        dst_template := "[ offset .. offset + length - 1 ]",
+    )
+);
+
+CapJitAddLogicTemplate(
+    rec(
+        variable_names := [ "length" ],
+        variable_filters := [ IsBigInt ],
+        src_template := "Length( [ 0 .. length - 1 ] )",
+        dst_template := "length",
+    )
+);
+
+CapJitAddLogicTemplate(
+    rec(
+        variable_names := [ "list", "index", "x" ],
+        src_template := "List( list, map -> ( i -> AsList( map )[1 + i] ) )[index]( x )",
+        dst_template := "List( list, AsList )[index][1 + x]",
+    )
+);
diff --git a/gap/SkeletalFinSets.gd b/gap/SkeletalFinSets.gd
index 6279abe..75a314d 100644
--- a/gap/SkeletalFinSets.gd
+++ b/gap/SkeletalFinSets.gd
@@ -29,6 +29,27 @@ DeclareCategory( "IsObjectInCategoryOfSkeletalFinSets",
 DeclareCategory( "IsMorphismInCategoryOfSkeletalFinSets",
                  IsCapCategoryMorphism );
 
+#! @Description
+#! The GAP category of categories
+#! of skeletal finite sets with morphisms given by functions.
+#! @Arguments object
+DeclareCategory( "IsCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions",
+                  IsCategoryOfSkeletalFinSets );
+
+#! @Description
+#! The GAP category of objects in the category
+#! of skeletal finite sets with morphisms given by functions.
+#! @Arguments object
+DeclareCategory( "IsObjectInCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions",
+                 IsObjectInCategoryOfSkeletalFinSets );
+
+#! @Description
+#! The GAP category of morphisms in the category
+#! of skeletal finite sets with morphisms given by functions.
+#! @Arguments object
+DeclareCategory( "IsMorphismInCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions",
+                 IsMorphismInCategoryOfSkeletalFinSets );
+
 #! @Section Attributes
 
 #! @Description
@@ -52,13 +73,23 @@ DeclareAttribute( "AsList",
 CapJitAddTypeSignature( "AsList", [ IsObjectInCategoryOfSkeletalFinSets ], rec( filter := IsList, element_type := rec( filter := IsBigInt ) ) );
 
 #! @Description
-#!  The graph defining the skeletal finite set morphism <A>phi</A>, see <Ref Oper="MapOfFinSets" Label="for IsObjectInCategoryOfSkeletalFinSets, IsList, IsObjectInCategoryOfSkeletalFinSets" />.
+#!  The function defining the skeletal finite set morphism <A>phi</A>, see <Ref Oper="MapOfFinSets" Label="for IsObjectInCategoryOfSkeletalFinSets, IsFunction, IsObjectInCategoryOfSkeletalFinSets" />.
 #! @Arguments phi
-#! @Returns a list
-DeclareAttribute( "AsList",
+#! @Returns a function
+DeclareAttribute( "AsFunc",
         IsMorphismInCategoryOfSkeletalFinSets );
 
-CapJitAddTypeSignature( "AsList", [ IsMorphismInCategoryOfSkeletalFinSets ], rec( filter := IsList, element_type := rec( filter := IsBigInt ) ) );
+BindGlobal( "SkeletalFinSets_func_type",
+        rec( filter := IsFunction, signature := [ [ rec( filter := IsBigInt ) ], rec( filter := IsBigInt ) ] ) );
+
+CapJitAddTypeSignature( "AsFunc", [ IsMorphismInCategoryOfSkeletalFinSets ], SkeletalFinSets_func_type );
+
+#! @Description
+#!  The list of images defining the skeletal finite set morphism <A>phi</A>, see <Ref Oper="MapOfFinSets" Label="for IsObjectInCategoryOfSkeletalFinSets, IsFunction, IsObjectInCategoryOfSkeletalFinSets" />.
+#! @Arguments phi
+#! @Returns a list
+DeclareAttribute( "ListOfImages",
+        IsMorphismInCategoryOfSkeletalFinSets );
 
 #! @Section Constructors
 
@@ -72,6 +103,16 @@ DeclareOperation( "CategoryOfSkeletalFinSets", [ ] );
 #!  It is automatically created while loading this package.
 DeclareGlobalName( "SkeletalFinSets" );
 
+#! @Description
+#!  Construct a category of skeletal finite sets with maps given by functions.
+#! @Returns a &CAP; category
+DeclareOperation( "CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions", [ ] );
+
+#! @Description
+#!  The default instance of the category of skeletal finite sets given by functions.
+#!  It is automatically created while loading this package.
+DeclareGlobalName( "SkeletalFinSetsWithMorphismsGivenByFunctions" );
+
 #! @Description
 #!  Construct a skeletal finite set residing in
 #!  the default instance of the category of skeletal finite sets <C>SkeletalFinSets</C>
@@ -92,13 +133,12 @@ KeyDependentOperation( "FinSet", IsCategoryOfSkeletalFinSets, IsBigInt, ReturnTr
 
 #! @Description
 #!  Construct a map $\phi:$<A>s</A>$\to$<A>t</A> of the skeletal finite sets <A>s</A> and <A>t</A>,
-#!  i.e., a morphism in the &CAP; category of <A>s</A>, where <A>G</A>
-#!  is a list of integers in <A>t</A> describing the graph of $\phi$.
-#! @Arguments s, G, t
+#!  i.e., a morphism in the &CAP; category of <A>s</A>, where <A>f</A>
+#!  is a function with values in <A>t</A> describing the graph of $\phi$.
+#! @Arguments s, f, t
 #! @Returns a &CAP; morphism
 DeclareOperation( "MapOfFinSets",
-        [ IsObjectInCategoryOfSkeletalFinSets, IsList, IsObjectInCategoryOfSkeletalFinSets ] );
-#! @InsertChunk  SkeletalMapOfFinSets
+        [ IsObjectInCategoryOfSkeletalFinSets, IsFunction, IsObjectInCategoryOfSkeletalFinSets ] );
 
 #! @Section Tools
 
diff --git a/gap/SkeletalFinSets.gi b/gap/SkeletalFinSets.gi
index 87331a6..3fea877 100644
--- a/gap/SkeletalFinSets.gi
+++ b/gap/SkeletalFinSets.gi
@@ -5,17 +5,21 @@
 #
 
 ##
-InstallMethod( CategoryOfSkeletalFinSets,
-               [ ],
-               
+InstallMethod( CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions,
+        "for no input",
+        [ ],
+        
   function ( )
     local cat;
     
-    cat := CreateCapCategoryWithDataTypes(
-        "SkeletalFinSets", IsCategoryOfSkeletalFinSets,
-        IsObjectInCategoryOfSkeletalFinSets, IsMorphismInCategoryOfSkeletalFinSets, IsCapCategoryTwoCell,
-        IsBigInt, rec( filter := IsList, element_type := rec( filter := IsBigInt ) ), fail
-    );
+    cat := CreateCapCategoryWithDataTypes( "SkeletalFinSets",
+                   IsCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions,
+                   IsObjectInCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions,
+                   IsMorphismInCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions,
+                   IsCapCategoryTwoCell,
+                   IsBigInt,
+                   SkeletalFinSets_func_type,
+                   fail );
     
     cat!.category_as_first_argument := true;
     
@@ -40,7 +44,7 @@ InstallMethod( CategoryOfSkeletalFinSets,
     
     if ValueOption( "no_precompiled_code" ) <> true then
         
-        ADD_FUNCTIONS_FOR_CategoryOfSkeletalFinSetsWithMorphismsGivenByListsPrecompiled( cat );
+        ADD_FUNCTIONS_FOR_CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled( cat );
         
     fi;
     
@@ -84,25 +88,59 @@ end );
 
 ## Morphisms
 
+##
+InstallOtherMethodForCompilerForCAP( ListOfImages,
+        "for a category of skeletal finite sets and a CAP map of skeletal finite sets",
+        [ IsCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions, IsMorphismInCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions ],
+        
+  function ( cat, map )
+    
+    return List( [ 0 .. Length( Source( map ) ) - 1 ], AsFunc( map ) );
+    
+end );
+
+##
+InstallMethod( AsList,
+        "for a CAP map of skeletal finite sets",
+        [ IsMorphismInCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions ],
+        
+  function ( map )
+    local L;
+    
+    L := ListOfImages( CapCategory( map ), map );
+    
+    if Length( L ) > 1 and First( L ) + 1 = L[2] and First( L ) < Last( L ) then
+        ConvertToRangeRep( L );
+    fi;
+    
+    return L;
+    
+end );
+
 ##
 InstallMethod( MapOfFinSets,
-        "for two CAP skeletal finite sets and a list",
-        [ IsObjectInCategoryOfSkeletalFinSets, IsList, IsObjectInCategoryOfSkeletalFinSets ],
+        "for two CAP skeletal finite sets and a function",
+        [ IsObjectInCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions, IsFunction, IsObjectInCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions ],
         
-  function ( s, G, t )
+  function ( s, f, t )
     
-    return MapOfFinSets( CapCategory( s ), s, G, t );
+    return MorphismConstructor( CapCategory( s ), s, f, t );
     
 end );
 
 ##
 InstallOtherMethod( MapOfFinSets,
-        "for a category of skeletal finite sets, two CAP skeletal finite sets and a list",
-        [ IsCategoryOfSkeletalFinSets, IsObjectInCategoryOfSkeletalFinSets, IsList, IsObjectInCategoryOfSkeletalFinSets ],
+        "for two CAP skeletal finite sets and a list",
+        [ IsObjectInCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions, IsList, IsObjectInCategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions ],
         
-  function ( cat, s, G, t )
+  function ( s, G, t )
+    local f;
     
-    return MorphismConstructor( cat, s, G, t );
+    f := CapJitTypedExpression(
+                 i -> G[1 + i],
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MapOfFinSets( s, f, t );
     
 end );
 
@@ -129,13 +167,13 @@ InstallMethod( Preimage,
         [ IsMorphismInCategoryOfSkeletalFinSets, IsList ],
         
   function ( phi, t )
-    local S;
+    local S, f;
     
     S := AsList( Source( phi ) );
     
-    phi := AsList( phi );
+    f := AsFunc( phi );
     
-    return Filtered( S, i -> phi[1 + i] in t );
+    return Filtered( S, i -> f( i ) in t );
     
 end );
 
@@ -159,7 +197,7 @@ InstallMethod( CallFuncList,
     
     x := L[1];
     
-    return AsList( phi )[1 + x];
+    return AsFunc( phi )( x );
     
 end );
 
@@ -264,13 +302,13 @@ end );
 
 ##
 AddMorphismConstructor( SkeletalFinSets,
-  function ( cat, source, images, range )
+  function ( cat, source, func, range )
     local map;
     
     map := CreateCapCategoryMorphismWithAttributes( cat,
-            source,
-            range,
-            AsList, images );
+                   source,
+                   range,
+                   AsFunc, func );
     
     #% CAP_JIT_DROP_NEXT_STATEMENT
     Assert( 4, IsWellDefined( map ) );
@@ -283,7 +321,7 @@ end );
 AddMorphismDatum( SkeletalFinSets,
   function ( cat, map )
     
-    return AsList( map );
+    return AsFunc( map );
     
 end );
 
@@ -302,20 +340,22 @@ end );
 ##
 AddIsWellDefinedForMorphisms( SkeletalFinSets,
   function ( cat, mor )
-    local s, rel, t;
+    local s, f, t;
     
     s := Length( Source( mor ) );
     
-    rel := AsList( mor );
+    f := AsFunc( mor );
     
     t := Length( Range( mor ) );
     
     ## For CompilerForCAP we need if-elif-else with the same structure
-    if not ForAll( rel, a -> IsBigInt( a ) and a >= 0 ) then
-        return false;
-    elif s <> Length( rel ) then
-        return false;
-    elif not ForAll( rel, a -> a < t ) then
+    if not ForAll( [ 0 .. s - 1 ],
+               function( x )
+                 local fx;
+                 fx := f( x );
+                 return IsBigInt( fx ) and fx >= 0 and fx < t;
+             end ) then
+             
         return false;
     else
         return true;
@@ -326,8 +366,12 @@ end );
 ##
 AddIsEqualForMorphisms( SkeletalFinSets,
   function ( cat, mor1, mor2 )
+    local f1, f2;
+    
+    f1 := AsFunc( mor1 );
+    f2 := AsFunc( mor2 );
     
-    return AsList( mor1 ) = AsList( mor2 );
+    return ForAll( [ 0 .. Length( Source( mor1 ) ) - 1 ], x -> f1( x ) = f2( x ) );
     
 end );
 
@@ -342,25 +386,32 @@ end );
 ##
 AddIdentityMorphism( SkeletalFinSets,
   function ( cat, n )
+    local f;
+    
+    f := CapJitTypedExpression(
+                 IdFunc,
+                 cat -> SkeletalFinSets_func_type );
     
-    return MorphismConstructor( cat, n, [ 0 .. Length( n ) - 1 ], n );
+    return MorphismConstructor( cat, n, f, n );
     
 end );
 
 ##
 AddPreCompose( SkeletalFinSets,
   function ( cat, map_pre, map_post )
-    local s, t, im_pre, im_post, cmp;
+    local s, t, pre, post, pre_post;
     
     s := Source( map_pre );
     t := Range( map_post );
     
-    im_pre := AsList( map_pre );
-    im_post := AsList( map_post );
+    pre := AsFunc( map_pre );
+    post := AsFunc( map_post );
     
-    cmp := List( s, i -> im_post[1 + im_pre[1 + i]] );
+    pre_post := CapJitTypedExpression(
+                        i -> post( pre( i ) ),
+                        cat -> SkeletalFinSets_func_type );
     
-    return MorphismConstructor( cat, s, cmp, t );
+    return MorphismConstructor( cat, s, pre_post, t );
     
 end );
 
@@ -368,7 +419,7 @@ end );
 AddImageObject( SkeletalFinSets,
   function ( cat, phi )
     
-    return ObjectConstructor( cat, BigInt( Length( Set( AsList( phi ) ) ) ) );
+    return ObjectConstructor( cat, BigInt( Length( Set( ListOfImages( cat, phi ) ) ) ) );
     
 end );
 
@@ -378,7 +429,7 @@ AddIsEpimorphism( SkeletalFinSets,
   function ( cat, phi )
     local imgs, t;
     
-    imgs := AsList( phi );
+    imgs := ListOfImages( cat, phi );
     
     t := Length( Range( phi ) );
     
@@ -399,7 +450,7 @@ AddIsMonomorphism( SkeletalFinSets,
   function ( cat, phi )
     local imgs, t;
     
-    imgs := AsList( phi );
+    imgs := ListOfImages( cat, phi );
     
     t := Length( Range( phi ) );
     
@@ -422,8 +473,8 @@ AddIsLiftable( SkeletalFinSets,
   function ( cat, f, g )
     local ff, gg, fff;
     
-    ff := AsList( f );
-    gg := AsList( g );
+    ff := ListOfImages( cat, f );
+    gg := ListOfImages( cat, g );
     
     if 100 * Length( ff ) < Length( gg ) then
         fff := Set( ff );
@@ -438,15 +489,19 @@ end );
 ##
 AddLift( SkeletalFinSets,
   function ( cat, f, g )
-    local S, T, gg, ff;
+    local S, T, gg, ff, chi;
     
     S := Source( f );
     T := Source( g );
     
-    gg := AsList( g );
-    ff := AsList( f );
+    gg := ListOfImages( cat, g );
+    ff := AsFunc( f );
     
-    return MorphismConstructor( cat, S, List( S, x -> -1 + BigInt( SafePosition( gg, ff[1 + x] ) ) ), T );
+    chi := CapJitTypedExpression(
+                   x -> -1 + BigInt( SafePosition( gg, ff( x ) ) ),
+                   cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, S, chi, T );
     
 end );
 
@@ -456,8 +511,8 @@ AddIsColiftable( SkeletalFinSets,
   function ( cat, f, g )
     local ff, gg;
     
-    ff := AsList( f );
-    gg := AsList( g );
+    ff := ListOfImages( cat, f );
+    gg := ListOfImages( cat, g );
     
     return ForAll( Set( ff ), i -> Length( Set( gg{Positions( ff, i )} ) ) = 1 );
     
@@ -466,15 +521,15 @@ end );
 ##
 AddColift( SkeletalFinSets,
   function ( cat, f, g )
-    local S, T, ff, gg, chi;
+    local S, T, ff, gg, c, chi;
     
     S := Range( f );
     T := Range( g );
     
-    ff := AsList( f );
-    gg := AsList( g );
+    ff := ListOfImages( cat, f );
+    gg := ListOfImages( cat, g );
     
-    chi :=
+    c :=
       function ( y )
         if not y in ff then
             return BigInt( 0 );
@@ -482,32 +537,45 @@ AddColift( SkeletalFinSets,
         return gg[SafePosition( ff, y )];
     end;
     
-    return MorphismConstructor( cat, S, List( S, y -> chi(y) ), T );
+    chi := CapJitTypedExpression(
+                   c,
+                   cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, S, chi, T );
     
 end );
 
 ##
 AddImageEmbeddingWithGivenImageObject( SkeletalFinSets,
   function ( cat, phi, image )
+    local images, f;
+    
+    images := Set( ListOfImages( cat, phi ) );
+    
+    f := CapJitTypedExpression(
+                 i -> images[1 + i],
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, image, f, Range( phi ) );
     
-    return MorphismConstructor( cat, image, Set( AsList( phi ) ), Range( phi ) );
-
 end );
 
 ##
 AddCoastrictionToImageWithGivenImageObject( SkeletalFinSets,
   function ( cat, phi, image_object )
-    local G, images, s, L, l, pi;
+    local G, images, s, f, l, pi;
     
-    G := AsList( phi );
+    G := ListOfImages( cat, phi );
     
     images := Set( G );
     
     s := Source( phi );
     
-    L := List( s, i -> -1 + BigInt( SafePosition( images, G[1 + i] ) ) );
+    f := CapJitTypedExpression(
+                 i -> -1 + BigInt( SafePosition( images, G[1 + i] ) ),
+                 cat -> SkeletalFinSets_func_type );
     
-    pi := MorphismConstructor( cat, s, L, image_object );
+    pi := MorphismConstructor( cat, s, f, image_object );
     
     #% CAP_JIT_DROP_NEXT_STATEMENT
     Assert( 3, IsEpimorphism( cat, pi ) );
@@ -538,8 +606,15 @@ end );
 ##
 AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject( SkeletalFinSets,
   function ( cat, m, t )
+    local zero, f;
+    
+    zero := BigInt( 0 );
     
-    return MorphismConstructor( cat, m, ListWithIdenticalEntries( Length( m ), BigInt( 0 ) ), t );
+    f := CapJitTypedExpression(
+                 i -> zero,
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, m, f, t );
     
 end );
 
@@ -554,7 +629,7 @@ end );
 ##
 AddProjectionInFactorOfDirectProductWithGivenDirectProduct( SkeletalFinSets,
   function ( cat, D, k, P )
-    local T, l, a;
+    local T, l, a, f;
     
     T := D[k];
     
@@ -562,14 +637,18 @@ AddProjectionInFactorOfDirectProductWithGivenDirectProduct( SkeletalFinSets,
     
     a := Product( List( D{[ 1 .. k - 1 ]}, Length ) );
     
-    return MorphismConstructor( cat, P, List( P, i -> RemInt( QuoInt( i, a ), l ) ), T );
+    f := CapJitTypedExpression(
+                 i -> RemInt( QuoInt( i, a ), l ),
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, P, f, T );
     
 end );
 
 ##
 AddUniversalMorphismIntoDirectProductWithGivenDirectProduct( SkeletalFinSets,
   function ( cat, D, T, tau, P )
-    local l, d, dd, taus;
+    local l, d, dd, taus, f;
     
     l := Length( D );
     
@@ -577,21 +656,25 @@ AddUniversalMorphismIntoDirectProductWithGivenDirectProduct( SkeletalFinSets,
     
     dd := List( [ 0 .. l - 1 ], j -> Product( d{[ 1 .. j ]} ) );
     
-    taus := List( tau, AsList );
+    taus := List( tau, AsFunc );
+    
+    f := CapJitTypedExpression(
+                 # if l = 0, then Sum( [ 0 .. l - 1 ], j -> ... ) = 0 ∈ TerminalObject = P
+                 i -> Sum( [ 0 .. l - 1 ], j -> dd[1 + j] * taus[1 + j]( i ) ),
+                 cat -> SkeletalFinSets_func_type );
     
-    # if l = 0, then Sum( [ 0 .. l - 1 ], j -> ... ) = 0 ∈ TerminalObject = P
-    return MorphismConstructor( cat, T, List( T, i -> Sum( [ 0 .. l - 1 ], j -> taus[1 + j][1 + i] * dd[1 + j] ) ), P );
+    return MorphismConstructor( cat, T, f, P );
     
 end );
 
 ##
 AddEqualizer( SkeletalFinSets,
   function ( cat, s, D )
-    local D2, Eq;
+    local funcs, Eq;
     
-    D2 := List( D, AsList );
+    funcs := List( D, AsFunc );
     
-    Eq := Filtered( [ 0 .. Length( s ) - 1 ], x -> ForAll( [ 1 .. Length( D ) - 1 ], j -> D2[j][1 + x] = D2[j + 1][1 + x] ) );
+    Eq := Filtered( [ 0 .. Length( s ) - 1 ], i -> ForAll( [ 1 .. Length( D ) - 1 ], j -> funcs[j]( i ) = funcs[j + 1]( i ) ) );
     
     return ObjectConstructor( cat, Length( Eq ) );
     
@@ -600,28 +683,36 @@ end );
 ##
 AddEmbeddingOfEqualizerWithGivenEqualizer( SkeletalFinSets,
   function ( cat, s, D, E )
-    local D2, Eq;
+    local funcs, Eq, f;
     
-    D2 := List( D, AsList );
+    funcs := List( D, AsFunc );
     
-    Eq := Filtered( [ 0 .. Length( s ) - 1 ], x -> ForAll( [ 1 .. Length( D ) - 1 ], j -> D2[j][1 + x] = D2[j + 1][1 + x] ) );
+    Eq := Filtered( [ 0 .. Length( s ) - 1 ], x -> ForAll( [ 1 .. Length( D ) - 1 ], j -> funcs[j]( x ) = funcs[j + 1]( x ) ) );
     
-    return MorphismConstructor( cat, E, Eq, s );
+    f := CapJitTypedExpression(
+                 i -> Eq[1 + i],
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, E, f, s );
     
 end );
 
 ##
 AddUniversalMorphismIntoEqualizerWithGivenEqualizer( SkeletalFinSets,
   function ( cat, s, D, test_object, tau, E )
-    local D2, Eq, t;
+    local funcs, Eq, t, f;
+    
+    funcs := List( D, AsFunc );
     
-    D2 := List( D, AsList );
+    Eq := Filtered( [ 0 .. Length( s ) - 1 ], x -> ForAll( [ 1 .. Length( D ) - 1 ], j -> funcs[j]( x ) = funcs[j + 1]( x ) ) );
     
-    Eq := Filtered( [ 0 .. Length( s ) - 1 ], x -> ForAll( [ 1 .. Length( D ) - 1 ], j -> D2[j][1 + x] = D2[j + 1][1 + x] ) );
+    t := AsFunc( tau );
     
-    t := AsList( tau );
+    f := CapJitTypedExpression(
+                 x -> -1 + BigInt( SafePosition( Eq, t( x ) ) ),
+                 cat -> SkeletalFinSets_func_type );
     
-    return MorphismConstructor( cat, test_object, List( test_object, x -> -1 + BigInt( SafePosition( Eq, t[1 + x] ) ) ), E );
+    return MorphismConstructor( cat, test_object, f, E );
     
 end );
 
@@ -647,8 +738,13 @@ end );
 ##
 AddUniversalMorphismFromInitialObjectWithGivenInitialObject( SkeletalFinSets,
   function ( cat, m, I )
+    local f;
     
-    return MorphismConstructor( cat, I, [ ], m );
+    f := CapJitTypedExpression(
+                 IdFunc,
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, I, f, m );
     
 end );
 
@@ -691,8 +787,13 @@ end );
 ##
 AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject( SkeletalFinSets,
   function ( cat, M, injective_object )
+    local f;
+    
+    f := CapJitTypedExpression(
+                 IdFunc,
+                 cat -> SkeletalFinSets_func_type );
     
-    return MorphismConstructor( cat, M, [ 0 .. Length( M ) - 1 ], injective_object );
+    return MorphismConstructor( cat, M, f, injective_object );
     
 end );
 
@@ -707,7 +808,7 @@ end );
 ##
 AddInjectionOfCofactorOfCoproductWithGivenCoproduct( SkeletalFinSets,
   function ( cat, L, i, coproduct )
-    local O, sum, s;
+    local O, sum, s, f;
     
     O := L{[ 1 .. i - 1 ]};
     
@@ -715,18 +816,41 @@ AddInjectionOfCofactorOfCoproductWithGivenCoproduct( SkeletalFinSets,
     
     s := L[i];
     
-    return MorphismConstructor( cat, s, [ sum .. sum + Length( s ) - 1 ], coproduct );
+    f := CapJitTypedExpression(
+                 i -> sum + i,
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, s, f, coproduct );
     
 end );
 
 ##
 AddUniversalMorphismFromCoproductWithGivenCoproduct( SkeletalFinSets,
   function ( cat, L, test_object, tau, S )
-    local concat;
+    local sources, sums, funcs, interval, c, f;
+    
+    sources := List( tau, map -> Length( Source( map ) ) );
+    
+    ## FIXME: make a functional loop if possible
+    sums := List( [ 0 .. Length( tau ) ], k -> Sum( sources{[ 1 .. k ]} ) );
     
-    concat := Concatenation( List( tau, AsList ) );
+    funcs := List( tau, AsFunc );
     
-    return MorphismConstructor( cat, S, concat, test_object );
+    interval := [ 1 .. Length( tau ) ];
+    
+    c :=
+      function( i )
+        local p;
+        ## FIXME: PositionSorted
+        p := SafePositionProperty( interval, k -> i < sums[k + 1] );
+        return funcs[p]( i - sums[p] );
+    end;
+    
+    f := CapJitTypedExpression(
+                 c,
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, S, f, test_object );
     
 end );
 
@@ -741,24 +865,32 @@ end );
 ##
 AddProjectionOntoCoequalizerWithGivenCoequalizer( SkeletalFinSets,
   function ( cat, s, D, C )
-    local Cq, cmp;
+    local Cq, f;
     
     Cq := SKELETAL_FIN_SETS_ExplicitCoequalizer( s, D );
     
-    cmp := List( s, x -> -1 + BigInt( SafeUniquePositionProperty( Cq, c -> x in c ) ) );
+    f := CapJitTypedExpression(
+                 i -> -1 + BigInt( SafeUniquePositionProperty( Cq, c -> i in c ) ),
+                 cat -> SkeletalFinSets_func_type );
     
-    return MorphismConstructor( cat, s, cmp, C );
+    return MorphismConstructor( cat, s, f, C );
     
 end );
 
 ##
 AddUniversalMorphismFromCoequalizerWithGivenCoequalizer( SkeletalFinSets,
   function ( cat, s, D, test_object, tau, C )
-    local Cq;
+    local Cq, func, f;
     
     Cq := SKELETAL_FIN_SETS_ExplicitCoequalizer( s, D );
 
-    return MorphismConstructor( cat, C, List( Cq, x -> tau( x[1] ) ), Range( tau ) );
+    func := AsFunc( tau );
+
+    f := CapJitTypedExpression(
+                 i -> func( Cq[1 + i][1] ),
+                 cat -> SkeletalFinSets_func_type );
+                 
+    return MorphismConstructor( cat, C, f, Range( tau ) );
     
 end );
 
@@ -799,13 +931,17 @@ end );
 ##
 AddCartesianBraidingInverseWithGivenDirectProducts( SkeletalFinSets,
   function ( cat, MN, M, N, NM )
-    local m, n;
+    local m, n, f;
     
     m := Length( M );
     
     n := Length( N );
     
-    return MorphismConstructor( cat, MN, List( MN , i -> RemInt( i, n ) * m + QuoInt( i, n ) ), NM );
+    f := CapJitTypedExpression(
+                 i -> RemInt( i, n ) * m + QuoInt( i, n ),
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, MN, f, NM );
     
 end );
 
@@ -824,35 +960,40 @@ end );
 ## InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism
 AddCartesianLambdaElimination( SkeletalFinSets,
   function ( cat, M, N, intro )
-    local m, n, v;
+    local m, n, v, f;
     
     m := Length( M );
     n := Length( N );
     
-    v := AsList( intro )[1];
+    v := AsFunc( intro )( 0 );
     
-    return MorphismConstructor( cat,
-                   M,
-                   List( [ 0 .. m - 1 ], i -> RemInt( QuoInt( v, n^i ), n ) ),
-                   N );
+    f := CapJitTypedExpression(
+                 i -> RemInt( QuoInt( v, n^i ), n ),
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, M, f, N );
     
 end );
 
 ## InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure
 AddCartesianLambdaIntroduction( SkeletalFinSets,
   function ( cat, map )
-    local M, m, N, n, images;
+    local M, m, N, n, func, f;
     
     M := Source( map );
     m := Length( M );
     N := Range( map );
     n := Length( N );
     
-    images := AsList( map );
+    func := AsFunc( map );
+    
+    f := CapJitTypedExpression(
+                 i -> Sum( List( [ 0 .. m - 1 ], k -> func( k ) * n^k ) ),
+                 cat -> SkeletalFinSets_func_type );
     
     return MorphismConstructor( cat,
                    TerminalObject( cat ),
-                   [ Sum( List( [ 0 .. m - 1 ], k -> images[1 + k] * n^k ) ) ],
+                   f,
                    ExponentialOnObjects( cat, M, N ) );
     
 end );
@@ -864,38 +1005,49 @@ AddExactCoverWithGlobalElements( SkeletalFinSets,
     
     T := TerminalObject( cat );
     
-    return List( [ 0 .. Length( A ) - 1 ], i -> MorphismConstructor( cat, T, [ i ], A ) );
+    return List( [ 0 .. Length( A ) - 1 ], j ->
+                 MorphismConstructor( cat,
+                         T,
+                         CapJitTypedExpression(
+                                 i -> j,
+                                 cat -> SkeletalFinSets_func_type ),
+                         A ) );
     
 end );
 
 ##
 AddExponentialOnMorphismsWithGivenExponentials( SkeletalFinSets,
   function ( cat, S, alpha, beta, T )
-    local M, N, MN, mors;
+    local M, N, D, MN, f;
     
     M := Range( alpha );
     N := Source( beta );
     
-    MN := ExponentialOnObjects( cat, M, N );
-
-    mors := ExactCoverWithGlobalElements( cat, MN );
+    D := TerminalObject( cat );
     
-    return MorphismConstructor( cat,
-                   S,
-                   List( mors, mor ->
-                         AsList( CartesianLambdaIntroduction( cat,
-                                 PreComposeList(
-                                         cat,
-                                         [ alpha,
-                                           CartesianLambdaElimination( cat,
-                                                   M,
-                                                   N,
-                                                   mor ),
-                                           beta ] ) ) )[1 + 0] ),
-                   T );
+    MN := ExponentialOnObjects( cat, M, N );
     
-end, 1 + Sum( [ [ "ExponentialOnObjects", 1 ],
-                [ "ExactCoverWithGlobalElements", 1 ],
+    f := CapJitTypedExpression(
+                 i -> AsFunc( CartesianLambdaIntroduction( cat,
+                         PreComposeList(
+                                 cat,
+                                 [ alpha,
+                                   CartesianLambdaElimination( cat,
+                                           M,
+                                           N,
+                                           MorphismConstructor( cat,
+                                                   D,
+                                                   CapJitTypedExpression(
+                                                           j -> i,
+                                                           cat -> SkeletalFinSets_func_type ),
+                                                   MN ) ),
+                                   beta ] ) ) )( 0 ),
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, S, f, T );
+    
+end, 1 + Sum( [ [ "TerminalObject", 1 ],
+                [ "ExponentialOnObjects", 1 ],
                 [ "PreComposeList", 2 ],
                 [ "CartesianLambdaElimination", 2 ],
                 [ "CartesianLambdaIntroduction", 2 ] ],
@@ -904,30 +1056,40 @@ end, 1 + Sum( [ [ "ExponentialOnObjects", 1 ],
 ##
 AddCartesianEvaluationMorphismWithGivenSource( SkeletalFinSets,
   function ( cat, M, N, HM_NxM )
-    local m, n, exp;
+    local m, n, exp, f;
     
     m := Length( M );
     n := Length( N );
     
     exp := n ^ m;
     
-    return MorphismConstructor( cat, HM_NxM, List( [ 0 .. ( n^m * m ) - 1 ], i -> RemInt( QuoInt( i, n^QuoInt( i, exp ) ), n ) ), N );
+    f := CapJitTypedExpression(
+                 i -> RemInt( QuoInt( i, n^QuoInt( i, exp ) ), n ),
+                 cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, HM_NxM, f, N );
     
 end );
 
 ##
 AddCartesianCoevaluationMorphismWithGivenRange( SkeletalFinSets,
   function ( cat, M, N, HN_MxN )
-    local m, n, mn;
+    local m, n, mn, f;
     
     m := Length( M );
     n := Length( N );
     
     mn := m * n;
     
-    #return MorphismConstructor( cat, M, List( [ 0 .. m - 1 ], i -> Sum( [ 0 .. n - 1 ], j -> ( i + m * j ) * (m*n)^j ) ), HN_MxN );
+    #f := CapJitTypedExpression(
+    #             i -> Sum( [ 0 .. n - 1 ], j -> ( i + m * j ) * (m*n)^j ),
+    #             cat -> SkeletalFinSets_func_type );
+    
+    f := CapJitTypedExpression(
+                 i -> i * GeometricSum( mn, n ) + m * mn * GeometricSumDiff1( mn, n ),
+                 cat -> SkeletalFinSets_func_type );
     
-    return MorphismConstructor( cat, M, List( [ 0 .. m - 1 ], i -> i * GeometricSum( mn, n ) + m * mn * GeometricSumDiff1( mn, n ) ), HN_MxN );
+    return MorphismConstructor( cat, M, f, HN_MxN );
     
 end );
 
@@ -942,25 +1104,28 @@ end );
 ##
 AddClassifyingMorphismOfSubobjectWithGivenSubobjectClassifier( SkeletalFinSets,
   function ( cat, monomorphism, Omega )
-    local range, images, chi;
+    local range, images, c, chi;
     
     range := Range( monomorphism );
     
-    images := AsList( monomorphism );
-    
-    chi := List( range,
-                 function ( x )
-                   
-                   if x in images then
-                       return BigInt( 1 );
-                   fi;
-                   
-                   return BigInt( 0 );
-                   
-               end );
-      
-      return MorphismConstructor( cat, range, chi, Omega );
-      
+    images := ListOfImages( cat, monomorphism );
+    
+    c :=
+      function ( x )
+        if x in images then
+            return BigInt( 1 );
+        fi;
+        
+        return BigInt( 0 );
+        
+    end;
+    
+    chi := CapJitTypedExpression(
+                   c,
+                   cat -> SkeletalFinSets_func_type );
+    
+    return MorphismConstructor( cat, range, chi, Omega );
+    
 end );
 
 ##
@@ -1105,14 +1270,3 @@ InstallMethod( DisplayString,
     return Concatenation( PrintString( phi ), "\n" );
     
 end );
-
-##
-InstallOtherMethod( FinSet,
-        "for a nonnegative integer",
-        [ IsBigInt ],
-        
-  function ( n )
-    
-    return FinSet( SkeletalFinSets, n );
-    
-end );
diff --git a/gap/SkeletalFinSetsEager.gd b/gap/SkeletalFinSetsEager.gd
new file mode 100644
index 0000000..785d3e3
--- /dev/null
+++ b/gap/SkeletalFinSetsEager.gd
@@ -0,0 +1,64 @@
+# SPDX-License-Identifier: GPL-2.0-or-later
+# FinSetsForCAP: The elementary topos of (skeletal) finite sets
+#
+# Declarations
+#
+
+#! @Chapter The category of skeletal finite sets with morphisms given by lists
+
+#! @Section GAP Categories
+
+#! @Description
+#! The GAP category of categories
+#! of skeletal finite sets with morphisms given by lists.
+#! @Arguments object
+DeclareCategory( "IsCategoryOfSkeletalFinSetsWithMorphismsGivenByLists",
+                  IsCategoryOfSkeletalFinSets );
+
+#! @Description
+#! The GAP category of objects in the category
+#! of skeletal finite sets with morphisms given by lists.
+#! @Arguments object
+DeclareCategory( "IsObjectInCategoryOfSkeletalFinSetsWithMorphismsGivenByLists",
+                 IsObjectInCategoryOfSkeletalFinSets );
+
+#! @Description
+#! The GAP category of morphisms in the category
+#! of skeletal finite sets with morphisms given by lists.
+#! @Arguments object
+DeclareCategory( "IsMorphismInCategoryOfSkeletalFinSetsWithMorphismsGivenByLists",
+                 IsMorphismInCategoryOfSkeletalFinSets );
+
+#! @Section Attributes
+
+#! @Description
+#!  The list of images defining the skeletal finite set morphism <A>phi</A>, see <Ref Oper="MapOfFinSets" Label="for IsObjectInCategoryOfSkeletalFinSets, IsFunction, IsObjectInCategoryOfSkeletalFinSets" />.
+#! @Arguments phi
+#! @Returns a list
+DeclareAttribute( "AsList",
+        IsMorphismInCategoryOfSkeletalFinSets );
+
+CapJitAddTypeSignature( "AsList", [ IsMorphismInCategoryOfSkeletalFinSetsWithMorphismsGivenByLists ],
+        rec( filter := IsList, element_type := rec( filter := IsBigInt ) ) );
+
+#! @Section Constructors
+
+#! @Description
+#!  Construct a category of skeletal finite sets with maps given by lists.
+#! @Returns a &CAP; category
+DeclareOperation( "CategoryOfSkeletalFinSetsWithMorphismsGivenByLists", [ ] );
+
+#! @Description
+#!  The default instance of the category of skeletal finite sets given by lists.
+#!  It is automatically created while loading this package.
+DeclareGlobalName( "SkeletalFinSetsWithMorphismsGivenByLists" );
+
+#! @Description
+#!  Construct a map $\phi:$<A>s</A>$\to$<A>t</A> of the skeletal finite sets <A>s</A> and <A>t</A>,
+#!  i.e., a morphism in the &CAP; category of <A>s</A>, where <A>list_of_images</A>
+#!  is a list of integers in <A>t</A> describing the images of $\phi$.
+#! @Arguments s, list_of_images, t
+#! @Returns a &CAP; morphism
+DeclareOperation( "MapOfFinSets",
+        [ IsObjectInCategoryOfSkeletalFinSets, IsList, IsObjectInCategoryOfSkeletalFinSets ] );
+#! @InsertChunk  SkeletalMapOfFinSets
diff --git a/gap/SkeletalFinSetsEager.gi b/gap/SkeletalFinSetsEager.gi
new file mode 100644
index 0000000..4427ab6
--- /dev/null
+++ b/gap/SkeletalFinSetsEager.gi
@@ -0,0 +1,282 @@
+# SPDX-License-Identifier: GPL-2.0-or-later
+# FinSetsForCAP: The elementary topos of (skeletal) finite sets
+#
+# Implementations
+#
+
+##
+InstallMethod( CategoryOfSkeletalFinSetsWithMorphismsGivenByLists,
+        "for no input",
+        [ ],
+        
+  function ( )
+    local object_constructor, object_datum,
+          morphism_constructor, morphism_datum,
+          skeletal_finsets_with_morphisms_given_by_functions,
+          modeling_tower_object_constructor, modeling_tower_object_datum,
+          modeling_tower_morphism_constructor, modeling_tower_morphism_datum,
+          skeletal_finsets_with_morphisms_given_by_lists;
+    
+    ##
+    object_constructor := function( cat, cardinality )
+        
+        return CreateCapCategoryObjectWithAttributes( cat,
+                       Length, cardinality );
+    
+    end;
+    
+    ##
+    object_datum := { cat, obj } ->  Length( obj );
+    
+    ##
+    morphism_constructor := function( cat, source, list_of_images, range )
+        
+        return CreateCapCategoryMorphismWithAttributes( cat,
+                       source,
+                       range,
+                       AsList, list_of_images );
+        
+    end;
+    
+    ##
+    morphism_datum := { cat, mor } -> AsList( mor );
+    
+    ## building the categorical tower:
+    skeletal_finsets_with_morphisms_given_by_functions :=
+      CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions( : FinalizeCategory := true );
+    
+    ## from the raw object data to the object in the modeling category
+    modeling_tower_object_constructor :=
+      function( cat, cardinality )
+        local skeletal_finsets_with_morphisms_given_by_functions;
+        
+        skeletal_finsets_with_morphisms_given_by_functions := ModelingCategory( cat );
+        
+        return ObjectConstructor( skeletal_finsets_with_morphisms_given_by_functions,
+                       cardinality );
+        
+    end;
+    
+    ## from the object in the modeling category to the raw object data
+    modeling_tower_object_datum := { cat, obj } -> Length( obj );
+    
+    ## from the raw morphism data to the morphism in the modeling category
+    modeling_tower_morphism_constructor :=
+      function( cat, source, list_of_images, range )
+        local skeletal_finsets_with_morphisms_given_by_functions, f;
+        
+        skeletal_finsets_with_morphisms_given_by_functions := ModelingCategory( cat );
+        
+        f := CapJitTypedExpression(
+                     i -> list_of_images[1 + i],
+                     cat -> SkeletalFinSets_func_type );
+        
+        return MorphismConstructor( skeletal_finsets_with_morphisms_given_by_functions,
+                       source,
+                       f,
+                       range );
+        
+    end;
+    
+    ## from the morphism in the modeling category to the raw morphism data
+    modeling_tower_morphism_datum :=
+      function( cat, mor )
+        
+        return List( [ 0 .. Length( Source( mor ) ) - 1 ], AsFunc( mor ) );
+        
+    end;
+    
+    skeletal_finsets_with_morphisms_given_by_lists :=
+      ReinterpretationOfCategory( skeletal_finsets_with_morphisms_given_by_functions,
+              rec( name := "SkeletalFinSets",
+                   category_filter := IsCategoryOfSkeletalFinSetsWithMorphismsGivenByLists,
+                   category_object_filter := IsObjectInCategoryOfSkeletalFinSetsWithMorphismsGivenByLists,
+                   category_morphism_filter := IsMorphismInCategoryOfSkeletalFinSetsWithMorphismsGivenByLists,
+                   object_constructor := object_constructor,
+                   object_datum := object_datum,
+                   morphism_constructor := morphism_constructor,
+                   morphism_datum := morphism_datum,
+                   modeling_tower_object_constructor := modeling_tower_object_constructor,
+                   modeling_tower_object_datum := modeling_tower_object_datum,
+                   modeling_tower_morphism_constructor := modeling_tower_morphism_constructor,
+                   modeling_tower_morphism_datum := modeling_tower_morphism_datum,
+                   only_primitive_operations := true,
+                   ) : FinalizeCategory := false );
+    
+    # this is a workhorse category -> no logic and caching only via IsIdenticalObj
+    CapCategorySwitchLogicOff( skeletal_finsets_with_morphisms_given_by_lists );
+    
+    SetRangeCategoryOfHomomorphismStructure( skeletal_finsets_with_morphisms_given_by_lists, skeletal_finsets_with_morphisms_given_by_lists );
+    SetIsEquippedWithHomomorphismStructure( skeletal_finsets_with_morphisms_given_by_lists, true );
+    
+    ##
+    AddIsEqualForMorphisms( skeletal_finsets_with_morphisms_given_by_lists,
+      function ( cat, mor1, mor2 )
+        
+        return AsList( mor1 ) = AsList( mor2 );
+        
+    end );
+    
+    ## the function SKELETAL_FIN_SETS_IsEpimorphism
+    ## has linear runtime complexity
+    AddIsEpimorphism( skeletal_finsets_with_morphisms_given_by_lists,
+      function ( cat, phi )
+        local imgs, t;
+        
+        imgs := AsList( phi );
+        
+        t := Length( Range( phi ) );
+        
+        ## we do not have a linear purely functional test (yet),
+        ## the following linear runtime function works with side effects,
+        ## so we hide it from the compiler
+        
+        return not (false in SKELETAL_FIN_SETS_IsEpimorphism( imgs, t ));
+        
+    end );
+    
+    ##
+    AddIsMonomorphism( skeletal_finsets_with_morphisms_given_by_lists,
+      function ( cat, phi )
+        local imgs, t;
+        
+        imgs := AsList( phi );
+        
+        t := Length( Range( phi ) );
+        
+        ## we do not have a linear purely functional test (yet),
+        ## the following linear runtime function works with side effects,
+        ## so we hide it from the compiler
+        
+        return SKELETAL_FIN_SETS_IsMonomorphism( imgs, t );
+        
+    end );
+    
+    ##
+    AddUniversalMorphismFromCoproductWithGivenCoproduct( skeletal_finsets_with_morphisms_given_by_lists,
+      function ( cat, L, test_object, tau, S )
+        local concat;
+        
+        concat := Concatenation( List( tau, AsList ) );
+        
+        return MorphismConstructor( cat, S, concat, test_object );
+        
+    end );
+    
+    ##
+    AddCoequalizer( skeletal_finsets_with_morphisms_given_by_lists,
+     function ( cat, s, D )
+        
+        return ObjectConstructor( cat, BigInt( Length( SKELETAL_FIN_SETS_ExplicitCoequalizer( s, D ) ) ) );
+        
+    end );
+    
+    ##
+    AddProjectionOntoCoequalizerWithGivenCoequalizer( skeletal_finsets_with_morphisms_given_by_lists,
+      function ( cat, s, D, C )
+        local Cq, cmp;
+        
+        Cq := SKELETAL_FIN_SETS_ExplicitCoequalizer( s, D );
+        
+        cmp := List( s, x -> -1 + BigInt( SafeUniquePositionProperty( Cq, c -> x in c ) ) );
+        
+        return MorphismConstructor( cat, s, cmp, C );
+        
+    end );
+    
+    ##
+    AddUniversalMorphismFromCoequalizerWithGivenCoequalizer( skeletal_finsets_with_morphisms_given_by_lists,
+      function ( cat, s, D, test_object, tau, C )
+        local Cq;
+        
+        Cq := SKELETAL_FIN_SETS_ExplicitCoequalizer( s, D );
+        
+        return MorphismConstructor( cat, C, List( Cq, x -> tau( x[1] ) ), Range( tau ) );
+        
+    end );
+    
+    ##
+    AddExactCoverWithGlobalElements( skeletal_finsets_with_morphisms_given_by_lists,
+      function ( cat, A )
+        local T;
+        
+        T := TerminalObject( cat );
+        
+        return List( [ 0 .. Length( A ) - 1 ], i ->
+                     MorphismConstructor( cat,
+                             T,
+                             [ i ],
+                             A ) );
+        
+    end );
+    
+    ##
+    AddMorphismsOfExternalHom( skeletal_finsets_with_morphisms_given_by_lists,
+      function ( cat, A, B )
+        local hom_A_B, mors;
+        
+        hom_A_B := ExponentialOnObjects( cat, A, B );
+        
+        mors := ExactCoverWithGlobalElements( cat, hom_A_B );
+        
+        return List( mors,
+                     mor -> CartesianLambdaElimination( cat,
+                             A,
+                             B,
+                             mor ) );
+        
+    end, 1 + Sum( [ [ "ExponentialOnObjects", 1 ],
+            [ "ExactCoverWithGlobalElements", 1 ],
+            [ "CartesianLambdaElimination", 2 ] ],
+            e -> e[2] * CurrentOperationWeight( skeletal_finsets_with_morphisms_given_by_lists!.derivations_weight_list, e[1] ) ) );
+    
+    if ValueOption( "no_precompiled_code" ) <> true then
+        
+        ADD_FUNCTIONS_FOR_CategoryOfSkeletalFinSetsWithMorphismsGivenByListsPrecompiled( skeletal_finsets_with_morphisms_given_by_lists );
+        
+    fi;
+    
+    Finalize( skeletal_finsets_with_morphisms_given_by_lists );
+    
+    return skeletal_finsets_with_morphisms_given_by_lists;
+    
+end );
+
+##
+InstallMethod( CategoryOfSkeletalFinSets, "for no input", [ ], CategoryOfSkeletalFinSetsWithMorphismsGivenByLists );
+
+##
+InstallOtherMethod( FinSet,
+        "for a nonnegative integer",
+        [ IsBigInt ],
+        
+  function ( n )
+    
+    return FinSet( SkeletalFinSets, n );
+    
+end );
+
+##
+InstallMethod( MapOfFinSets,
+        "for two CAP skeletal finite sets and a list",
+        [ IsObjectInCategoryOfSkeletalFinSetsWithMorphismsGivenByLists, IsList, IsObjectInCategoryOfSkeletalFinSetsWithMorphismsGivenByLists ],
+        
+  function ( s, list_of_images, t )
+    
+    return MorphismConstructor( CapCategory( s ), s, list_of_images, t );
+    
+end );
+
+##
+InstallMethod( AsFunc,
+        "for a CAP map of skeletal finite sets",
+        [ IsMorphismInCategoryOfSkeletalFinSetsWithMorphismsGivenByLists ],
+        
+  function ( map )
+    local images;
+    
+    images := AsList( map );
+    
+    return i -> images[1 + i];
+    
+end );
diff --git a/gap/init.gi b/gap/init.gi
index 333d2e5..b2f33d1 100644
--- a/gap/init.gi
+++ b/gap/init.gi
@@ -10,7 +10,13 @@ BindGlobal( "init_FinSetsForCAP", function ( )
     BindGlobal( "FinSets", CategoryOfFinSets( ) );
     
     ##
-    BindGlobal( "SkeletalFinSets", CategoryOfSkeletalFinSets( ) );
+    BindGlobal( "SkeletalFinSetsWithMorphismsGivenByFunctions", CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions( ) );
+    
+    ##
+    BindGlobal( "SkeletalFinSetsWithMorphismsGivenByLists", CategoryOfSkeletalFinSetsWithMorphismsGivenByLists( ) );
+    
+    ##
+    BindGlobal( "SkeletalFinSets", SkeletalFinSetsWithMorphismsGivenByLists );
     
 end );
 
diff --git a/gap/precompiled_categories/CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled.gi b/gap/precompiled_categories/CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled.gi
new file mode 100644
index 0000000..77ced12
--- /dev/null
+++ b/gap/precompiled_categories/CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled.gi
@@ -0,0 +1,1016 @@
+# SPDX-License-Identifier: GPL-2.0-or-later
+# FinSetsForCAP: The elementary topos of (skeletal) finite sets
+#
+# Implementations
+#
+BindGlobal( "ADD_FUNCTIONS_FOR_CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled", function ( cat )
+    
+    ##
+    AddCartesianBraidingInverseWithGivenDirectProducts( cat,
+        
+########
+function ( cat_1, s_1, a_1, b_1, r_1 )
+    local deduped_1_1, hoisted_2_1;
+    hoisted_2_1 := Length( a_1 );
+    deduped_1_1 := Length( b_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, s_1, r_1, AsFunc, function ( i_2 )
+            return REM_INT( i_2, deduped_1_1 ) * hoisted_2_1 + QUO_INT( i_2, deduped_1_1 );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCartesianCoevaluationMorphismWithGivenRange( cat,
+        
+########
+function ( cat_1, a_1, b_1, r_1 )
+    local hoisted_1_1, hoisted_2_1, deduped_3_1, deduped_4_1, deduped_5_1;
+    deduped_5_1 := Length( b_1 );
+    deduped_4_1 := Length( a_1 );
+    deduped_3_1 := deduped_4_1 * deduped_5_1;
+    hoisted_2_1 := deduped_4_1 * deduped_3_1 * GeometricSumDiff1( deduped_3_1, deduped_5_1 );
+    hoisted_1_1 := GeometricSum( deduped_3_1, deduped_5_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, r_1, AsFunc, function ( i_2 )
+            return i_2 * hoisted_1_1 + hoisted_2_1;
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCartesianEvaluationMorphismWithGivenSource( cat,
+        
+########
+function ( cat_1, a_1, b_1, s_1 )
+    local hoisted_1_1, deduped_3_1;
+    deduped_3_1 := Length( b_1 );
+    hoisted_1_1 := deduped_3_1 ^ Length( a_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, s_1, b_1, AsFunc, function ( i_2 )
+            return REM_INT( QUO_INT( i_2, deduped_3_1 ^ QUO_INT( i_2, hoisted_1_1 ) ), deduped_3_1 );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCartesianLambdaElimination( cat,
+        
+########
+function ( cat_1, a_1, b_1, alpha_1 )
+    local deduped_1_1, hoisted_2_1;
+    hoisted_2_1 := AsFunc( alpha_1 )( 0 );
+    deduped_1_1 := Length( b_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, b_1, AsFunc, function ( i_2 )
+            return REM_INT( QUO_INT( hoisted_2_1, deduped_1_1 ^ i_2 ), deduped_1_1 );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCartesianLambdaIntroduction( cat,
+        
+########
+function ( cat_1, alpha_1 )
+    local hoisted_1_1, hoisted_3_1, deduped_4_1, deduped_5_1;
+    deduped_5_1 := Length( Source( alpha_1 ) );
+    deduped_4_1 := Length( Range( alpha_1 ) );
+    hoisted_1_1 := AsFunc( alpha_1 );
+    hoisted_3_1 := Sum( List( [ 0 .. deduped_5_1 - 1 ], function ( k_2 )
+              return hoisted_1_1( k_2 ) * deduped_4_1 ^ k_2;
+          end ) );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( 1 ) ), CreateCapCategoryObjectWithAttributes( cat_1, Length, deduped_4_1 ^ deduped_5_1 ), AsFunc, function ( i_2 )
+            return hoisted_3_1;
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCartesianLeftUnitorInverseWithGivenDirectProduct( cat,
+        
+########
+function ( cat_1, a_1, r_1 )
+    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, a_1, AsFunc, ID_FUNC );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCartesianLeftUnitorWithGivenDirectProduct( cat,
+        
+########
+function ( cat_1, a_1, s_1 )
+    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, a_1, AsFunc, ID_FUNC );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCartesianRightUnitorInverseWithGivenDirectProduct( cat,
+        
+########
+function ( cat_1, a_1, r_1 )
+    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, a_1, AsFunc, ID_FUNC );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCartesianRightUnitorWithGivenDirectProduct( cat,
+        
+########
+function ( cat_1, a_1, s_1 )
+    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, a_1, AsFunc, ID_FUNC );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddClassifyingMorphismOfSubobjectWithGivenSubobjectClassifier( cat,
+        
+########
+function ( cat_1, alpha_1, Omega_1 )
+    local hoisted_1_1, hoisted_2_1;
+    hoisted_2_1 := BigInt( 1 );
+    hoisted_1_1 := List( [ 0 .. Length( Source( alpha_1 ) ) - 1 ], AsFunc( alpha_1 ) );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, Range( alpha_1 ), Omega_1, AsFunc, function ( x_2 )
+            if x_2 in hoisted_1_1 then
+                return hoisted_2_1;
+            else
+                return BigInt( 0 );
+            fi;
+            return;
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCoastrictionToImageWithGivenImageObject( cat,
+        
+########
+function ( cat_1, alpha_1, I_1 )
+    local hoisted_2_1, deduped_3_1, deduped_4_1;
+    deduped_4_1 := Source( alpha_1 );
+    deduped_3_1 := List( [ 0 .. Length( deduped_4_1 ) - 1 ], AsFunc( alpha_1 ) );
+    hoisted_2_1 := SSortedList( deduped_3_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, deduped_4_1, I_1, AsFunc, function ( i_2 )
+            return -1 + BigInt( SafePosition( hoisted_2_1, deduped_3_1[(1 + i_2)] ) );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCoequalizer( cat,
+        
+########
+function ( cat_1, arg2_1, arg3_1 )
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( Length( SKELETAL_FIN_SETS_ExplicitCoequalizer( arg2_1, arg3_1 ) ) ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddColift( cat,
+        
+########
+function ( cat_1, alpha_1, beta_1 )
+    local deduped_1_1, hoisted_2_1, hoisted_3_1;
+    hoisted_3_1 := List( [ 0 .. Length( Source( beta_1 ) ) - 1 ], AsFunc( beta_1 ) );
+    hoisted_2_1 := BigInt( 0 );
+    deduped_1_1 := List( [ 0 .. Length( Source( alpha_1 ) ) - 1 ], AsFunc( alpha_1 ) );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, Range( alpha_1 ), Range( beta_1 ), AsFunc, function ( y_2 )
+            if not y_2 in deduped_1_1 then
+                return hoisted_2_1;
+            else
+                return hoisted_3_1[SafePosition( deduped_1_1, y_2 )];
+            fi;
+            return;
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddCoproduct( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, Sum( List( arg2_1, Length ) ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddDirectProduct( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, Product( List( arg2_1, Length ) ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddEmbeddingOfEqualizerWithGivenEqualizer( cat,
+        
+########
+function ( cat_1, Y_1, morphisms_1, P_1 )
+    local deduped_1_1, hoisted_2_1, hoisted_3_1;
+    hoisted_2_1 := [ 1 .. Length( morphisms_1 ) - 1 ];
+    deduped_1_1 := List( morphisms_1, AsFunc );
+    hoisted_3_1 := Filtered( [ 0 .. Length( Y_1 ) - 1 ], function ( x_2 )
+            return ForAll( hoisted_2_1, function ( j_3 )
+                    return deduped_1_1[j_3]( x_2 ) = deduped_1_1[j_3 + 1]( x_2 );
+                end );
+        end );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, P_1, Y_1, AsFunc, function ( i_2 )
+            return hoisted_3_1[1 + i_2];
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddEpimorphismFromSomeProjectiveObject( cat,
+        
+########
+function ( cat_1, A_1 )
+    return CreateCapCategoryMorphismWithAttributes( cat_1, A_1, A_1, AsFunc, ID_FUNC );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddEqualizer( cat,
+        
+########
+function ( cat_1, arg2_1, arg3_1 )
+    local deduped_1_1, hoisted_2_1;
+    hoisted_2_1 := [ 1 .. Length( arg3_1 ) - 1 ];
+    deduped_1_1 := List( arg3_1, AsFunc );
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, Length( Filtered( [ 0 .. Length( arg2_1 ) - 1 ], function ( i_2 )
+                return ForAll( hoisted_2_1, function ( j_3 )
+                        return deduped_1_1[j_3]( i_2 ) = deduped_1_1[j_3 + 1]( i_2 );
+                    end );
+            end ) ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddExactCoverWithGlobalElements( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    local hoisted_1_1;
+    hoisted_1_1 := CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( 1 ) );
+    return List( [ 0 .. Length( arg2_1 ) - 1 ], function ( j_2 )
+            return CreateCapCategoryMorphismWithAttributes( cat_1, hoisted_1_1, arg2_1, AsFunc, function ( i_3 )
+                    return j_2;
+                end );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddExponentialOnMorphismsWithGivenExponentials( cat,
+        
+########
+function ( cat_1, s_1, alpha_1, beta_1, r_1 )
+    local hoisted_1_1, deduped_2_1, hoisted_3_1, hoisted_4_1, hoisted_5_1;
+    hoisted_5_1 := [ 0 .. Length( Source( alpha_1 ) ) - 1 ];
+    hoisted_4_1 := Length( Range( beta_1 ) );
+    hoisted_3_1 := AsFunc( beta_1 );
+    deduped_2_1 := Length( Source( beta_1 ) );
+    hoisted_1_1 := AsFunc( alpha_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, s_1, r_1, AsFunc, function ( i_2 )
+            return Sum( List( hoisted_5_1, function ( k_3 )
+                      return hoisted_3_1( REM_INT( QUO_INT( i_2, deduped_2_1 ^ hoisted_1_1( k_3 ) ), deduped_2_1 ) ) * hoisted_4_1 ^ k_3;
+                  end ) );
+        end );
+end
+########
+        
+    , 1003 );
+    
+    ##
+    AddExponentialOnObjects( cat,
+        
+########
+function ( cat_1, a_1, b_1 )
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, Length( b_1 ) ^ Length( a_1 ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIdentityMorphism( cat,
+        
+########
+function ( cat_1, a_1 )
+    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, a_1, AsFunc, ID_FUNC );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddImageEmbeddingWithGivenImageObject( cat,
+        
+########
+function ( cat_1, alpha_1, I_1 )
+    local hoisted_1_1;
+    hoisted_1_1 := SSortedList( List( [ 0 .. Length( Source( alpha_1 ) ) - 1 ], AsFunc( alpha_1 ) ) );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, I_1, Range( alpha_1 ), AsFunc, function ( i_2 )
+            return hoisted_1_1[1 + i_2];
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddImageObject( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( Length( SSortedList( List( [ 0 .. Length( Source( arg2_1 ) ) - 1 ], AsFunc( arg2_1 ) ) ) ) ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddInitialObject( cat,
+        
+########
+function ( cat_1 )
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( 0 ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddInjectionOfCofactorOfCoproductWithGivenCoproduct( cat,
+        
+########
+function ( cat_1, objects_1, k_1, P_1 )
+    local hoisted_1_1;
+    hoisted_1_1 := Sum( List( objects_1, Length ){[ 1 .. k_1 - 1 ]} );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, objects_1[k_1], P_1, AsFunc, function ( i_2 )
+            return hoisted_1_1 + i_2;
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsColiftable( cat,
+        
+########
+function ( cat_1, arg2_1, arg3_1 )
+    local hoisted_2_1, deduped_3_1;
+    deduped_3_1 := List( [ 0 .. Length( Source( arg2_1 ) ) - 1 ], AsFunc( arg2_1 ) );
+    hoisted_2_1 := List( [ 0 .. Length( Source( arg3_1 ) ) - 1 ], AsFunc( arg3_1 ) );
+    return ForAll( SSortedList( deduped_3_1 ), function ( i_2 )
+            return Length( SSortedList( hoisted_2_1{Positions( deduped_3_1, i_2 )} ) ) = 1;
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsEpimorphism( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return not false in SKELETAL_FIN_SETS_IsEpimorphism( List( [ 0 .. Length( Source( arg2_1 ) ) - 1 ], AsFunc( arg2_1 ) ), Length( Range( arg2_1 ) ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsEqualForMorphisms( cat,
+        
+########
+function ( cat_1, arg2_1, arg3_1 )
+    local hoisted_1_1, hoisted_2_1;
+    hoisted_2_1 := AsFunc( arg3_1 );
+    hoisted_1_1 := AsFunc( arg2_1 );
+    return ForAll( [ 0 .. Length( Source( arg2_1 ) ) - 1 ], function ( x_2 )
+            return hoisted_1_1( x_2 ) = hoisted_2_1( x_2 );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsEqualForObjects( cat,
+        
+########
+function ( cat_1, arg2_1, arg3_1 )
+    return Length( arg2_1 ) = Length( arg3_1 );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsHomSetInhabited( cat,
+        
+########
+function ( cat_1, arg2_1, arg3_1 )
+    return Length( arg2_1 ) = 0 or Length( arg3_1 ) <> 0;
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsInitial( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return Length( arg2_1 ) = 0;
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsInjective( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return Length( arg2_1 ) <> 0;
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsLiftable( cat,
+        
+########
+function ( cat_1, arg2_1, arg3_1 )
+    local hoisted_1_1, deduped_2_1, deduped_3_1, deduped_4_1;
+    deduped_4_1 := [ 0 .. Length( Source( arg3_1 ) ) - 1 ];
+    deduped_3_1 := [ 0 .. Length( Source( arg2_1 ) ) - 1 ];
+    deduped_2_1 := List( deduped_3_1, AsFunc( arg2_1 ) );
+    hoisted_1_1 := List( deduped_4_1, AsFunc( arg3_1 ) );
+    return ForAll( IdFunc( function (  )
+                if 100 * Length( deduped_3_1 ) < Length( deduped_4_1 ) then
+                    return SSortedList( deduped_2_1 );
+                else
+                    return deduped_2_1;
+                fi;
+                return;
+            end )(  ), function ( y_2 )
+            return y_2 in hoisted_1_1;
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsMonomorphism( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return SKELETAL_FIN_SETS_IsMonomorphism( List( [ 0 .. Length( Source( arg2_1 ) ) - 1 ], AsFunc( arg2_1 ) ), Length( Range( arg2_1 ) ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsProjective( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return true;
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsSplitEpimorphism( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return not false in SKELETAL_FIN_SETS_IsEpimorphism( List( [ 0 .. Length( Source( arg2_1 ) ) - 1 ], AsFunc( arg2_1 ) ), Length( Range( arg2_1 ) ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsSplitMonomorphism( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    local deduped_1_1, deduped_2_1;
+    deduped_2_1 := Length( Source( arg2_1 ) );
+    deduped_1_1 := Length( Range( arg2_1 ) );
+    return deduped_1_1 = 0 or deduped_2_1 <> 0 and SKELETAL_FIN_SETS_IsMonomorphism( List( [ 0 .. deduped_2_1 - 1 ], AsFunc( arg2_1 ) ), deduped_1_1 );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsTerminal( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return Length( arg2_1 ) = 1;
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsWellDefinedForMorphisms( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    local hoisted_1_1, hoisted_2_1;
+    hoisted_2_1 := Length( Range( arg2_1 ) );
+    hoisted_1_1 := AsFunc( arg2_1 );
+    if not ForAll( [ 0 .. Length( Source( arg2_1 ) ) - 1 ], function ( x_2 )
+                 local deduped_1_2;
+                 deduped_1_2 := hoisted_1_1( x_2 );
+                 return (IsBigInt( deduped_1_2 ) and deduped_1_2 >= 0 and deduped_1_2 < hoisted_2_1);
+             end ) then
+        return false;
+    else
+        return true;
+    fi;
+    return;
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddIsWellDefinedForObjects( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return Length( arg2_1 ) >= 0;
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddLift( cat,
+        
+########
+function ( cat_1, alpha_1, beta_1 )
+    local hoisted_1_1, hoisted_2_1, deduped_3_1;
+    deduped_3_1 := Source( beta_1 );
+    hoisted_2_1 := List( [ 0 .. Length( deduped_3_1 ) - 1 ], AsFunc( beta_1 ) );
+    hoisted_1_1 := AsFunc( alpha_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, Source( alpha_1 ), deduped_3_1, AsFunc, function ( x_2 )
+            return -1 + BigInt( SafePosition( hoisted_2_1, hoisted_1_1( x_2 ) ) );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject( cat,
+        
+########
+function ( cat_1, A_1, I_1 )
+    return CreateCapCategoryMorphismWithAttributes( cat_1, A_1, I_1, AsFunc, ID_FUNC );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddMorphismConstructor( cat,
+        
+########
+function ( cat_1, arg2_1, arg3_1, arg4_1 )
+    return CreateCapCategoryMorphismWithAttributes( cat_1, arg2_1, arg4_1, AsFunc, arg3_1 );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddMorphismDatum( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return AsFunc( arg2_1 );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddMorphismsOfExternalHom( cat,
+        
+########
+function ( cat_1, arg2_1, arg3_1 )
+    local deduped_2_1;
+    deduped_2_1 := Length( arg3_1 );
+    return List( [ 0 .. deduped_2_1 ^ Length( arg2_1 ) - 1 ], function ( logic_new_func_x_2 )
+            return CreateCapCategoryMorphismWithAttributes( cat_1, arg2_1, arg3_1, AsFunc, function ( i_3 )
+                    return REM_INT( QUO_INT( logic_new_func_x_2, deduped_2_1 ^ i_3 ), deduped_2_1 );
+                end );
+        end );
+end
+########
+        
+    , 401 );
+    
+    ##
+    AddObjectConstructor( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, arg2_1 );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddObjectDatum( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    return Length( arg2_1 );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddPreCompose( cat,
+        
+########
+function ( cat_1, alpha_1, beta_1 )
+    local hoisted_1_1, hoisted_2_1;
+    hoisted_2_1 := AsFunc( beta_1 );
+    hoisted_1_1 := AsFunc( alpha_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, Source( alpha_1 ), Range( beta_1 ), AsFunc, function ( i_2 )
+            return hoisted_2_1( hoisted_1_1( i_2 ) );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddProjectionInFactorOfDirectProductWithGivenDirectProduct( cat,
+        
+########
+function ( cat_1, objects_1, k_1, P_1 )
+    local hoisted_1_1, hoisted_2_1, deduped_3_1;
+    deduped_3_1 := List( objects_1, Length );
+    hoisted_2_1 := deduped_3_1[k_1];
+    hoisted_1_1 := Product( deduped_3_1{[ 1 .. k_1 - 1 ]} );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, P_1, objects_1[k_1], AsFunc, function ( i_2 )
+            return REM_INT( QUO_INT( i_2, hoisted_1_1 ), hoisted_2_1 );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddProjectionOntoCoequalizerWithGivenCoequalizer( cat,
+        
+########
+function ( cat_1, Y_1, morphisms_1, P_1 )
+    local hoisted_1_1;
+    hoisted_1_1 := SKELETAL_FIN_SETS_ExplicitCoequalizer( Y_1, morphisms_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, Y_1, P_1, AsFunc, function ( i_2 )
+            return -1 + BigInt( SafeUniquePositionProperty( hoisted_1_1, function ( c_3 )
+                        return (i_2 in c_3);
+                    end ) );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddPushoutComplement( cat,
+        
+########
+function ( cat_1, l_1, m_1 )
+    local deduped_5_1, deduped_6_1, hoisted_8_1, hoisted_10_1, hoisted_15_1, hoisted_18_1, hoisted_19_1, deduped_22_1, deduped_23_1, deduped_24_1, deduped_25_1, deduped_26_1, deduped_27_1, deduped_28_1, deduped_29_1, deduped_30_1, deduped_31_1, deduped_32_1, deduped_33_1, deduped_34_1, deduped_35_1, deduped_36_1, deduped_37_1, deduped_38_1;
+    deduped_38_1 := BigInt( 3 );
+    deduped_37_1 := BigInt( 2 );
+    deduped_36_1 := BigInt( 1 );
+    deduped_35_1 := BigInt( 0 );
+    deduped_34_1 := Range( m_1 );
+    deduped_33_1 := Length( Source( l_1 ) );
+    deduped_32_1 := Length( Source( m_1 ) );
+    deduped_31_1 := Length( deduped_34_1 );
+    deduped_30_1 := Filtered( [ deduped_35_1, deduped_36_1, deduped_37_1, deduped_38_1 ], function ( x_2 )
+            return IdFunc( function (  )
+                        if (x_2 = deduped_38_1) then
+                            return deduped_36_1;
+                        else
+                            return deduped_35_1;
+                        fi;
+                        return;
+                    end )(  ) = REM_INT( x_2, deduped_37_1 );
+        end );
+    hoisted_10_1 := AsFunc( l_1 );
+    deduped_29_1 := Filtered( [ 0 .. deduped_32_1 * deduped_33_1 - 1 ], function ( i_2 )
+            return REM_INT( QUO_INT( i_2, deduped_36_1 ), deduped_32_1 ) = hoisted_10_1( REM_INT( QUO_INT( i_2, deduped_32_1 ), deduped_33_1 ) );
+        end );
+    deduped_28_1 := Filtered( [ 0 .. deduped_32_1 * deduped_32_1 - 1 ], function ( x_2 )
+            return REM_INT( QUO_INT( x_2, deduped_36_1 ), deduped_32_1 ) = REM_INT( QUO_INT( x_2, deduped_32_1 ), deduped_32_1 );
+        end );
+    deduped_27_1 := Length( deduped_29_1 );
+    deduped_26_1 := Filtered( [ 0 .. deduped_27_1 * deduped_33_1 - 1 ], function ( x_2 )
+            return REM_INT( QUO_INT( deduped_29_1[1 + REM_INT( QUO_INT( x_2, deduped_36_1 ), deduped_27_1 )], deduped_32_1 ), deduped_33_1 ) = REM_INT( QUO_INT( x_2, deduped_27_1 ), deduped_33_1 );
+        end );
+    deduped_6_1 := deduped_31_1 * deduped_35_1;
+    deduped_5_1 := AsFunc( m_1 );
+    deduped_25_1 := SSortedList( List( [ 0 .. Length( deduped_28_1 ) - 1 ], function ( i_2 )
+              return deduped_36_1 * deduped_5_1( REM_INT( QUO_INT( deduped_28_1[(1 + i_2)], deduped_36_1 ), deduped_32_1 ) ) + deduped_6_1;
+          end ) );
+    deduped_23_1 := SSortedList( List( [ 0 .. Length( deduped_26_1 ) - 1 ], function ( i_2 )
+              return deduped_36_1 * deduped_5_1( REM_INT( QUO_INT( deduped_29_1[(1 + REM_INT( QUO_INT( deduped_26_1[(1 + i_2)], deduped_36_1 ), deduped_27_1 ))], deduped_36_1 ), deduped_32_1 ) ) + deduped_6_1;
+          end ) );
+    hoisted_19_1 := [ deduped_35_1 ];
+    hoisted_18_1 := deduped_30_1{[ 1 .. Length( deduped_30_1 ) ]};
+    hoisted_15_1 := List( [ 0 .. BigInt( Length( deduped_23_1 ) ) - 1 ], function ( i_2 )
+            return deduped_23_1[1 + i_2];
+        end );
+    hoisted_8_1 := List( [ 0 .. BigInt( Length( deduped_25_1 ) ) - 1 ], function ( i_2 )
+            return deduped_25_1[1 + i_2];
+        end );
+    deduped_24_1 := Filtered( [ 0 .. deduped_31_1 - 1 ], function ( i_2 )
+            local deduped_1_2;
+            deduped_1_2 := REM_INT( QUO_INT( i_2, deduped_36_1 ), deduped_31_1 );
+            return IdFunc( function (  )
+                        if (deduped_36_1 * IdFunc( function (  )
+                                          if (deduped_1_2 in hoisted_8_1) then
+                                              return deduped_36_1;
+                                          else
+                                              return deduped_35_1;
+                                          fi;
+                                          return;
+                                      end )(  ) + deduped_37_1 * IdFunc( function (  )
+                                          if (deduped_1_2 in hoisted_15_1) then
+                                              return deduped_36_1;
+                                          else
+                                              return deduped_35_1;
+                                          fi;
+                                          return;
+                                      end )(  ) in hoisted_18_1) then
+                            return deduped_36_1;
+                        else
+                            return deduped_35_1;
+                        fi;
+                        return;
+                    end )(  ) = IdFunc( function (  )
+                        if (REM_INT( QUO_INT( i_2, deduped_31_1 ), deduped_36_1 ) in hoisted_19_1) then
+                            return deduped_36_1;
+                        else
+                            return deduped_35_1;
+                        fi;
+                        return;
+                    end )(  );
+        end );
+    deduped_22_1 := SSortedList( List( [ 0 .. Length( deduped_24_1 ) - 1 ], function ( i_2 )
+              return REM_INT( QUO_INT( deduped_24_1[1 + i_2], deduped_36_1 ), deduped_31_1 );
+          end ) );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( Length( deduped_22_1 ) ) ), deduped_34_1, AsFunc, function ( i_2 )
+            return deduped_22_1[1 + i_2];
+        end );
+end
+########
+        
+    , 19767 : IsPrecompiledDerivation := true );
+    
+    ##
+    AddSomeInjectiveObject( cat,
+        
+########
+function ( cat_1, arg2_1 )
+    if Length( arg2_1 ) = 0 then
+        return CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( 1 ) );
+    else
+        return arg2_1;
+    fi;
+    return;
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddSubobjectClassifier( cat,
+        
+########
+function ( cat_1 )
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( 2 ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddTerminalObject( cat,
+        
+########
+function ( cat_1 )
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( 1 ) );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddUniversalMorphismFromCoequalizerWithGivenCoequalizer( cat,
+        
+########
+function ( cat_1, Y_1, morphisms_1, T_1, tau_1, P_1 )
+    local hoisted_1_1, hoisted_2_1;
+    hoisted_2_1 := AsFunc( tau_1 );
+    hoisted_1_1 := SKELETAL_FIN_SETS_ExplicitCoequalizer( Y_1, morphisms_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, P_1, Range( tau_1 ), AsFunc, function ( i_2 )
+            return hoisted_2_1( hoisted_1_1[1 + i_2][1] );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddUniversalMorphismFromCoproductWithGivenCoproduct( cat,
+        
+########
+function ( cat_1, objects_1, T_1, tau_1, P_1 )
+    local hoisted_1_1, deduped_2_1, hoisted_3_1, hoisted_4_1, deduped_5_1;
+    deduped_5_1 := Length( tau_1 );
+    hoisted_4_1 := List( tau_1, AsFunc );
+    hoisted_3_1 := [ 1 .. deduped_5_1 ];
+    hoisted_1_1 := List( tau_1, function ( map_2 )
+            return Length( Source( map_2 ) );
+        end );
+    deduped_2_1 := List( [ 0 .. deduped_5_1 ], function ( k_2 )
+            return Sum( hoisted_1_1{[ 1 .. k_2 ]} );
+        end );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, P_1, T_1, AsFunc, function ( i_2 )
+            local deduped_1_2;
+            deduped_1_2 := SafePositionProperty( hoisted_3_1, function ( k_3 )
+                    return i_2 < deduped_2_1[k_3 + 1];
+                end );
+            return hoisted_4_1[deduped_1_2]( i_2 - deduped_2_1[deduped_1_2] );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddUniversalMorphismFromInitialObjectWithGivenInitialObject( cat,
+        
+########
+function ( cat_1, T_1, P_1 )
+    return CreateCapCategoryMorphismWithAttributes( cat_1, P_1, T_1, AsFunc, ID_FUNC );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddUniversalMorphismIntoDirectProductWithGivenDirectProduct( cat,
+        
+########
+function ( cat_1, objects_1, T_1, tau_1, P_1 )
+    local hoisted_1_1, hoisted_2_1, hoisted_3_1, deduped_5_1;
+    deduped_5_1 := [ 0 .. Length( objects_1 ) - 1 ];
+    hoisted_3_1 := List( tau_1, AsFunc );
+    hoisted_1_1 := List( objects_1, Length );
+    hoisted_2_1 := List( deduped_5_1, function ( j_2 )
+            return Product( hoisted_1_1{[ 1 .. j_2 ]} );
+        end );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, T_1, P_1, AsFunc, function ( i_2 )
+            return Sum( deduped_5_1, function ( j_3 )
+                    local deduped_1_3;
+                    deduped_1_3 := 1 + j_3;
+                    return hoisted_2_1[deduped_1_3] * hoisted_3_1[deduped_1_3]( i_2 );
+                end );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddUniversalMorphismIntoEqualizerWithGivenEqualizer( cat,
+        
+########
+function ( cat_1, Y_1, morphisms_1, T_1, tau_1, P_1 )
+    local deduped_1_1, hoisted_2_1, hoisted_3_1, hoisted_4_1;
+    hoisted_2_1 := [ 1 .. Length( morphisms_1 ) - 1 ];
+    deduped_1_1 := List( morphisms_1, AsFunc );
+    hoisted_4_1 := Filtered( [ 0 .. Length( Y_1 ) - 1 ], function ( x_2 )
+            return ForAll( hoisted_2_1, function ( j_3 )
+                    return deduped_1_1[j_3]( x_2 ) = deduped_1_1[j_3 + 1]( x_2 );
+                end );
+        end );
+    hoisted_3_1 := AsFunc( tau_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, T_1, P_1, AsFunc, function ( x_2 )
+            return -1 + BigInt( SafePosition( hoisted_4_1, hoisted_3_1( x_2 ) ) );
+        end );
+end
+########
+        
+    , 100 );
+    
+    ##
+    AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject( cat,
+        
+########
+function ( cat_1, T_1, P_1 )
+    local hoisted_1_1;
+    hoisted_1_1 := BigInt( 0 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, T_1, P_1, AsFunc, function ( i_2 )
+            return hoisted_1_1;
+        end );
+end
+########
+        
+    , 100 );
+    
+    if IsBound( cat!.precompiled_functions_added ) then
+        
+        # COVERAGE_IGNORE_NEXT_LINE
+        Error( "precompiled functions have already been added before" );
+        
+    fi;
+    
+    cat!.precompiled_functions_added := true;
+    
+end );
+
+BindGlobal( "CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled", function (  )
+  local category_constructor, cat;
+    
+    category_constructor :=
+        
+        
+        function (  )
+    return CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctions(  );
+end;
+        
+        
+    
+    cat := category_constructor(  : FinalizeCategory := false, no_precompiled_code := true );
+    
+    ADD_FUNCTIONS_FOR_CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled( cat );
+    
+    Finalize( cat );
+    
+    return cat;
+    
+end );
diff --git a/gap/precompiled_categories/CategoryOfSkeletalFinSetsWithMorphismsGivenByListsPrecompiled.gi b/gap/precompiled_categories/CategoryOfSkeletalFinSetsWithMorphismsGivenByListsPrecompiled.gi
index 9fb24ec..57679d6 100644
--- a/gap/precompiled_categories/CategoryOfSkeletalFinSetsWithMorphismsGivenByListsPrecompiled.gi
+++ b/gap/precompiled_categories/CategoryOfSkeletalFinSetsWithMorphismsGivenByListsPrecompiled.gi
@@ -45,12 +45,11 @@ end
         
 ########
 function ( cat_1, a_1, b_1, s_1 )
-    local deduped_3_1, deduped_4_1, deduped_5_1;
-    deduped_5_1 := Length( a_1 );
-    deduped_4_1 := Length( b_1 );
-    deduped_3_1 := deduped_4_1 ^ deduped_5_1;
-    return CreateCapCategoryMorphismWithAttributes( cat_1, s_1, b_1, AsList, List( [ 0 .. deduped_3_1 * deduped_5_1 - 1 ], function ( i_2 )
-              return REM_INT( QUO_INT( i_2, deduped_4_1 ^ QUO_INT( i_2, deduped_3_1 ) ), deduped_4_1 );
+    local hoisted_1_1, deduped_3_1;
+    deduped_3_1 := Length( b_1 );
+    hoisted_1_1 := deduped_3_1 ^ Length( a_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, s_1, b_1, AsList, List( [ 0 .. Length( s_1 ) - 1 ], function ( i_2 )
+              return REM_INT( QUO_INT( i_2, deduped_3_1 ^ QUO_INT( i_2, hoisted_1_1 ) ), deduped_3_1 );
           end ) );
 end
 ########
@@ -95,7 +94,7 @@ end
         
 ########
 function ( cat_1, a_1, r_1 )
-    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, a_1, AsList, [ 0 .. Length( a_1 ) - 1 ] );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, r_1, AsList, [ 0 .. Length( a_1 ) - 1 ] );
 end
 ########
         
@@ -106,7 +105,7 @@ end
         
 ########
 function ( cat_1, a_1, s_1 )
-    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, a_1, AsList, [ 0 .. Length( a_1 ) - 1 ] );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, s_1, a_1, AsList, [ 0 .. Length( a_1 ) - 1 ] );
 end
 ########
         
@@ -117,7 +116,7 @@ end
         
 ########
 function ( cat_1, a_1, r_1 )
-    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, a_1, AsList, [ 0 .. Length( a_1 ) - 1 ] );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, r_1, AsList, [ 0 .. Length( a_1 ) - 1 ] );
 end
 ########
         
@@ -128,7 +127,7 @@ end
         
 ########
 function ( cat_1, a_1, s_1 )
-    return CreateCapCategoryMorphismWithAttributes( cat_1, a_1, a_1, AsList, [ 0 .. Length( a_1 ) - 1 ] );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, s_1, a_1, AsList, [ 0 .. Length( a_1 ) - 1 ] );
 end
 ########
         
@@ -238,12 +237,12 @@ function ( cat_1, Y_1, morphisms_1, P_1 )
     hoisted_2_1 := [ 1 .. Length( morphisms_1 ) - 1 ];
     deduped_1_1 := List( morphisms_1, AsList );
     return CreateCapCategoryMorphismWithAttributes( cat_1, P_1, Y_1, AsList, Filtered( [ 0 .. Length( Y_1 ) - 1 ], function ( x_2 )
-              local deduped_1_2;
-              deduped_1_2 := 1 + x_2;
-              return ForAll( hoisted_2_1, function ( j_3 )
-                      return deduped_1_1[j_3][deduped_1_2] = deduped_1_1[j_3 + 1][deduped_1_2];
-                  end );
-          end ) );
+                local deduped_1_2;
+                deduped_1_2 := 1 + x_2;
+                return ForAll( hoisted_2_1, function ( j_3 )
+                        return deduped_1_1[j_3][deduped_1_2] = deduped_1_1[j_3 + 1][deduped_1_2];
+                    end );
+            end ){[ 1 .. Length( P_1 ) ]} );
 end
 ########
         
@@ -268,9 +267,9 @@ function ( cat_1, arg2_1, arg3_1 )
     local deduped_1_1, hoisted_2_1;
     hoisted_2_1 := [ 1 .. Length( arg3_1 ) - 1 ];
     deduped_1_1 := List( arg3_1, AsList );
-    return CreateCapCategoryObjectWithAttributes( cat_1, Length, Length( Filtered( [ 0 .. Length( arg2_1 ) - 1 ], function ( x_2 )
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, Length( Filtered( [ 0 .. Length( arg2_1 ) - 1 ], function ( i_2 )
                 local deduped_1_2;
-                deduped_1_2 := 1 + x_2;
+                deduped_1_2 := 1 + i_2;
                 return ForAll( hoisted_2_1, function ( j_3 )
                         return deduped_1_1[j_3][deduped_1_2] = deduped_1_1[j_3 + 1][deduped_1_2];
                     end );
@@ -300,27 +299,21 @@ end
         
 ########
 function ( cat_1, s_1, alpha_1, beta_1, r_1 )
-    local hoisted_2_1, hoisted_3_1, hoisted_4_1, hoisted_5_1, hoisted_6_1, deduped_7_1, deduped_8_1;
-    deduped_8_1 := Length( Range( alpha_1 ) );
-    deduped_7_1 := Length( Source( beta_1 ) );
-    hoisted_6_1 := [ 0 .. Length( Source( alpha_1 ) ) - 1 ];
-    hoisted_5_1 := Length( Range( beta_1 ) );
-    hoisted_4_1 := AsList( beta_1 );
-    hoisted_3_1 := AsList( alpha_1 );
-    hoisted_2_1 := [ 0 .. deduped_8_1 - 1 ];
-    return CreateCapCategoryMorphismWithAttributes( cat_1, s_1, r_1, AsList, List( [ 0 .. deduped_7_1 ^ deduped_8_1 - 1 ], function ( i_2 )
-              local hoisted_1_2;
-              hoisted_1_2 := List( hoisted_2_1, function ( i_3 )
-                      return REM_INT( QUO_INT( i_2, deduped_7_1 ^ i_3 ), deduped_7_1 );
-                  end );
-              return Sum( List( hoisted_6_1, function ( k_3 )
-                        return hoisted_4_1[(1 + hoisted_1_2[(1 + hoisted_3_1[(1 + CAP_JIT_INCOMPLETE_LOGIC( k_3 ))])])] * hoisted_5_1 ^ k_3;
+    local hoisted_1_1, deduped_2_1, hoisted_3_1, hoisted_4_1, hoisted_5_1;
+    hoisted_5_1 := [ 0 .. Length( Source( alpha_1 ) ) - 1 ];
+    hoisted_4_1 := Length( Range( beta_1 ) );
+    hoisted_3_1 := AsList( beta_1 );
+    deduped_2_1 := Length( Source( beta_1 ) );
+    hoisted_1_1 := AsList( alpha_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, s_1, r_1, AsList, List( [ 0 .. Length( s_1 ) - 1 ], function ( i_2 )
+              return Sum( List( hoisted_5_1, function ( k_3 )
+                        return hoisted_3_1[(1 + REM_INT( QUO_INT( i_2, deduped_2_1 ^ hoisted_1_1[(1 + k_3)] ), deduped_2_1 ))] * hoisted_4_1 ^ k_3;
                     end ) );
           end ) );
 end
 ########
         
-    , 1003 );
+    , 100 );
     
     ##
     AddExponentialOnObjects( cat,
@@ -349,7 +342,7 @@ end
         
 ########
 function ( cat_1, alpha_1, I_1 )
-    return CreateCapCategoryMorphismWithAttributes( cat_1, I_1, Range( alpha_1 ), AsList, SSortedList( AsList( alpha_1 ) ) );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, I_1, Range( alpha_1 ), AsList, SSortedList( AsList( alpha_1 ) ){[ 1 .. Length( I_1 ) ]} );
 end
 ########
         
@@ -478,18 +471,18 @@ end
         
 ########
 function ( cat_1, arg2_1, arg3_1 )
-    local deduped_2_1, deduped_3_1;
-    deduped_3_1 := AsList( arg3_1 );
+    local hoisted_1_1, deduped_2_1;
     deduped_2_1 := AsList( arg2_1 );
+    hoisted_1_1 := AsList( arg3_1 );
     return ForAll( IdFunc( function (  )
-                if 100 * Length( deduped_2_1 ) < Length( deduped_3_1 ) then
+                if 100 * Length( Source( arg2_1 ) ) < Length( Source( arg3_1 ) ) then
                     return SSortedList( deduped_2_1 );
                 else
                     return deduped_2_1;
                 fi;
                 return;
             end )(  ), function ( y_2 )
-            return y_2 in deduped_3_1;
+            return y_2 in hoisted_1_1;
         end );
 end
 ########
@@ -558,17 +551,15 @@ end
         
 ########
 function ( cat_1, arg2_1 )
-    local hoisted_1_1, deduped_2_1;
-    deduped_2_1 := AsList( arg2_1 );
-    hoisted_1_1 := Length( Range( arg2_1 ) );
-    if not ForAll( deduped_2_1, function ( a_2 )
-                 return (IsBigInt( a_2 ) and a_2 >= 0);
-             end ) then
-        return false;
-    elif Length( Source( arg2_1 ) ) <> Length( deduped_2_1 ) then
-        return false;
-    elif not ForAll( deduped_2_1, function ( a_2 )
-                 return a_2 < hoisted_1_1;
+    local hoisted_1_1, hoisted_3_1, deduped_4_1;
+    deduped_4_1 := AsList( arg2_1 );
+    hoisted_3_1 := Length( Range( arg2_1 ) );
+    hoisted_1_1 := List( deduped_4_1, IsBigInt );
+    if not ForAll( [ 0 .. Length( Source( arg2_1 ) ) - 1 ], function ( x_2 )
+                 local deduped_1_2, deduped_2_2;
+                 deduped_2_2 := 1 + x_2;
+                 deduped_1_2 := deduped_4_1[deduped_2_2];
+                 return (hoisted_1_1[deduped_2_2] and deduped_1_2 >= 0 and deduped_1_2 < hoisted_3_1);
              end ) then
         return false;
     else
@@ -738,77 +729,137 @@ end
         
 ########
 function ( cat_1, l_1, m_1 )
-    local deduped_3_1, deduped_5_1, hoisted_6_1, deduped_8_1, hoisted_10_1, hoisted_13_1, hoisted_15_1, hoisted_16_1, hoisted_17_1, hoisted_18_1, deduped_20_1, deduped_21_1, deduped_22_1, deduped_23_1, deduped_24_1, deduped_25_1, deduped_26_1, deduped_27_1, deduped_28_1, deduped_29_1, deduped_30_1, deduped_31_1, deduped_32_1, deduped_33_1, deduped_34_1, deduped_35_1, deduped_36_1;
-    deduped_36_1 := BigInt( 1 );
-    deduped_35_1 := BigInt( 0 );
-    deduped_34_1 := Range( m_1 );
-    deduped_33_1 := Length( Source( l_1 ) );
-    deduped_32_1 := Length( Source( m_1 ) );
-    deduped_31_1 := Length( deduped_34_1 );
-    deduped_30_1 := [ 0 .. deduped_31_1 - 1 ];
-    deduped_29_1 := [ 0 .. deduped_32_1 * deduped_33_1 - 1 ];
-    deduped_28_1 := [ 0 .. deduped_32_1 * deduped_32_1 - 1 ];
-    deduped_3_1 := List( deduped_28_1, function ( i_2 )
-            return REM_INT( i_2, deduped_32_1 );
-        end );
-    deduped_27_1 := Filtered( deduped_28_1, function ( x_2 )
-            return deduped_3_1[1 + x_2] = REM_INT( QUO_INT( CAP_JIT_INCOMPLETE_LOGIC( x_2 ), deduped_32_1 ), deduped_32_1 );
+    local deduped_3_1, hoisted_4_1, hoisted_5_1, deduped_6_1, deduped_7_1, hoisted_8_1, deduped_10_1, hoisted_12_1, deduped_13_1, hoisted_15_1, hoisted_16_1, hoisted_17_1, hoisted_18_1, hoisted_19_1, hoisted_20_1, hoisted_22_1, hoisted_23_1, deduped_24_1, hoisted_25_1, deduped_26_1, deduped_27_1, deduped_28_1, deduped_29_1, deduped_30_1, deduped_31_1, deduped_32_1, deduped_33_1, deduped_34_1, deduped_35_1, deduped_36_1, deduped_37_1, deduped_38_1, deduped_39_1, deduped_40_1, deduped_41_1, deduped_42_1, deduped_43_1, deduped_44_1, deduped_45_1, deduped_46_1, deduped_47_1, deduped_48_1, deduped_49_1, deduped_50_1, deduped_51_1, deduped_52_1, deduped_53_1, deduped_54_1, deduped_55_1, deduped_56_1, deduped_57_1;
+    deduped_57_1 := BigInt( 3 );
+    deduped_56_1 := BigInt( 2 );
+    deduped_55_1 := BigInt( 1 );
+    deduped_54_1 := BigInt( 0 );
+    deduped_53_1 := Range( m_1 );
+    deduped_52_1 := Length( Source( l_1 ) );
+    deduped_51_1 := Length( Source( m_1 ) );
+    deduped_50_1 := Length( deduped_53_1 );
+    deduped_49_1 := [ deduped_54_1, deduped_56_1, deduped_57_1 ]{[ 1, 2, 3 ]};
+    deduped_48_1 := [ 0 .. deduped_50_1 - 1 ];
+    deduped_47_1 := [ 0 .. deduped_51_1 * deduped_52_1 - 1 ];
+    deduped_46_1 := [ 0 .. deduped_51_1 * deduped_51_1 - 1 ];
+    deduped_3_1 := List( deduped_46_1, function ( i_2 )
+            return REM_INT( i_2, deduped_51_1 );
         end );
-    hoisted_10_1 := AsList( l_1 );
-    deduped_8_1 := List( deduped_29_1, function ( i_2 )
-            return REM_INT( i_2, deduped_32_1 );
+    deduped_45_1 := Filtered( deduped_46_1, function ( x_2 )
+            return deduped_3_1[1 + x_2] = REM_INT( QUO_INT( CAP_JIT_INCOMPLETE_LOGIC( x_2 ), deduped_51_1 ), deduped_51_1 );
         end );
-    deduped_26_1 := Filtered( deduped_29_1, function ( x_2 )
-            return deduped_8_1[1 + x_2] = hoisted_10_1[1 + REM_INT( QUO_INT( CAP_JIT_INCOMPLETE_LOGIC( x_2 ), deduped_32_1 ), deduped_33_1 )];
+    deduped_44_1 := Length( deduped_45_1 );
+    deduped_43_1 := [ 1 .. deduped_44_1 ];
+    hoisted_12_1 := AsList( l_1 );
+    deduped_10_1 := List( deduped_47_1, function ( i_2 )
+            return REM_INT( i_2, deduped_51_1 );
         end );
-    deduped_25_1 := Length( deduped_26_1 );
-    deduped_24_1 := [ 0 .. deduped_25_1 - 1 ];
-    hoisted_13_1 := List( deduped_24_1, function ( i_2 )
-            return REM_INT( QUO_INT( CAP_JIT_INCOMPLETE_LOGIC( deduped_26_1[1 + i_2] ), deduped_32_1 ), deduped_33_1 );
+    deduped_42_1 := Filtered( deduped_47_1, function ( x_2 )
+            return deduped_10_1[1 + x_2] = hoisted_12_1[1 + REM_INT( QUO_INT( CAP_JIT_INCOMPLETE_LOGIC( x_2 ), deduped_51_1 ), deduped_52_1 )];
         end );
-    deduped_23_1 := Filtered( [ 0 .. deduped_25_1 * deduped_33_1 - 1 ], function ( x_2 )
+    deduped_41_1 := [ 0 .. deduped_44_1 - 1 ];
+    deduped_40_1 := Length( deduped_42_1 );
+    deduped_39_1 := [ 1 .. deduped_40_1 ];
+    deduped_38_1 := [ 0 .. deduped_40_1 - 1 ];
+    deduped_7_1 := deduped_50_1 * deduped_54_1;
+    deduped_6_1 := AsList( m_1 );
+    hoisted_4_1 := deduped_45_1{deduped_43_1};
+    hoisted_5_1 := List( deduped_41_1, function ( i_2 )
+              return deduped_3_1[1 + hoisted_4_1[(1 + i_2)]];
+          end ){deduped_43_1};
+    deduped_37_1 := SSortedList( List( deduped_41_1, function ( i_2 )
+              return deduped_6_1[1 + hoisted_5_1[(1 + CAP_JIT_INCOMPLETE_LOGIC( i_2 ))]] + deduped_7_1;
+          end ) );
+    deduped_13_1 := deduped_42_1{deduped_39_1};
+    hoisted_15_1 := (List( deduped_38_1, function ( i_2 )
+                return REM_INT( QUO_INT( CAP_JIT_INCOMPLETE_LOGIC( deduped_13_1[1 + i_2] ), deduped_51_1 ), deduped_52_1 );
+            end ){deduped_39_1}){deduped_39_1};
+    deduped_36_1 := Filtered( [ 0 .. deduped_40_1 * deduped_52_1 - 1 ], function ( x_2 )
             local deduped_1_2;
             deduped_1_2 := CAP_JIT_INCOMPLETE_LOGIC( x_2 );
-            return hoisted_13_1[1 + REM_INT( deduped_1_2, deduped_25_1 )] = REM_INT( QUO_INT( deduped_1_2, deduped_25_1 ), deduped_33_1 );
-        end );
-    hoisted_18_1 := [ deduped_36_1, deduped_35_1, deduped_36_1, deduped_36_1 ];
-    hoisted_17_1 := BigInt( 2 );
-    deduped_5_1 := AsList( m_1 );
-    hoisted_15_1 := List( deduped_24_1, function ( i_2 )
-            return deduped_5_1[1 + deduped_8_1[(1 + deduped_26_1[(1 + i_2)])]];
+            return hoisted_15_1[1 + REM_INT( deduped_1_2, deduped_40_1 )] = REM_INT( QUO_INT( deduped_1_2, deduped_40_1 ), deduped_52_1 );
         end );
-    hoisted_16_1 := List( [ 0 .. Length( deduped_23_1 ) - 1 ], function ( i_2 )
-            return hoisted_15_1[1 + REM_INT( CAP_JIT_INCOMPLETE_LOGIC( deduped_23_1[(1 + i_2)] ), deduped_25_1 )];
+    deduped_34_1 := Length( deduped_36_1 );
+    deduped_33_1 := [ 1 .. deduped_34_1 ];
+    deduped_30_1 := [ 0 .. deduped_34_1 - 1 ];
+    hoisted_16_1 := List( deduped_38_1, function ( i_2 )
+              return deduped_10_1[1 + deduped_13_1[(1 + i_2)]];
+          end ){deduped_39_1};
+    hoisted_19_1 := List( deduped_38_1, function ( i_2 )
+            return deduped_6_1[1 + hoisted_16_1[(1 + i_2)]];
         end );
-    hoisted_6_1 := List( [ 0 .. Length( deduped_27_1 ) - 1 ], function ( i_2 )
-            return deduped_5_1[1 + deduped_3_1[(1 + deduped_27_1[(1 + i_2)])]];
-        end );
-    deduped_22_1 := Filtered( deduped_30_1, function ( x_2 )
+    hoisted_17_1 := deduped_36_1{deduped_33_1};
+    hoisted_18_1 := List( deduped_30_1, function ( i_2 )
+              return REM_INT( CAP_JIT_INCOMPLETE_LOGIC( hoisted_17_1[1 + i_2] ), deduped_40_1 );
+          end ){deduped_33_1};
+    deduped_28_1 := SSortedList( List( deduped_30_1, function ( i_2 )
+              return hoisted_19_1[1 + hoisted_18_1[(1 + CAP_JIT_INCOMPLETE_LOGIC( i_2 ))]] + deduped_7_1;
+          end ) );
+    hoisted_22_1 := [ IdFunc( function (  )
+                  if deduped_54_1 in deduped_49_1 then
+                      return deduped_55_1;
+                  else
+                      return deduped_54_1;
+                  fi;
+                  return;
+              end )(  ), IdFunc( function (  )
+                  if deduped_55_1 in deduped_49_1 then
+                      return deduped_55_1;
+                  else
+                      return deduped_54_1;
+                  fi;
+                  return;
+              end )(  ), IdFunc( function (  )
+                  if deduped_56_1 in deduped_49_1 then
+                      return deduped_55_1;
+                  else
+                      return deduped_54_1;
+                  fi;
+                  return;
+              end )(  ), IdFunc( function (  )
+                  if deduped_57_1 in deduped_49_1 then
+                      return deduped_55_1;
+                  else
+                      return deduped_54_1;
+                  fi;
+                  return;
+              end )(  ) ];
+    hoisted_20_1 := deduped_28_1{[ 1 .. BigInt( Length( deduped_28_1 ) ) ]};
+    hoisted_8_1 := deduped_37_1{[ 1 .. BigInt( Length( deduped_37_1 ) ) ]};
+    deduped_35_1 := Filtered( deduped_48_1, function ( i_2 )
             local deduped_1_2;
-            deduped_1_2 := CAP_JIT_INCOMPLETE_LOGIC( REM_INT( CAP_JIT_INCOMPLETE_LOGIC( x_2 ), deduped_31_1 ) );
-            return hoisted_18_1[1 + (IdFunc( function (  )
-                             if (deduped_1_2 in hoisted_6_1) then
-                                 return deduped_36_1;
+            deduped_1_2 := CAP_JIT_INCOMPLETE_LOGIC( REM_INT( CAP_JIT_INCOMPLETE_LOGIC( i_2 ), deduped_50_1 ) );
+            return hoisted_22_1[1 + (IdFunc( function (  )
+                             if (deduped_1_2 in hoisted_8_1) then
+                                 return deduped_55_1;
                              else
-                                 return deduped_35_1;
+                                 return deduped_54_1;
                              fi;
                              return;
-                         end )(  ) + IdFunc( function (  )
-                               if (deduped_1_2 in hoisted_16_1) then
-                                   return deduped_36_1;
+                         end )(  ) + deduped_56_1 * IdFunc( function (  )
+                               if (deduped_1_2 in hoisted_20_1) then
+                                   return deduped_55_1;
                                else
-                                   return deduped_35_1;
+                                   return deduped_54_1;
                                fi;
                                return;
-                           end )(  ) * hoisted_17_1)] = deduped_36_1;
+                           end )(  ))] = deduped_55_1;
         end );
-    deduped_20_1 := List( deduped_30_1, function ( i_2 )
-            return REM_INT( i_2, deduped_31_1 );
+    deduped_32_1 := Length( deduped_35_1 );
+    deduped_31_1 := [ 1 .. deduped_32_1 ];
+    deduped_29_1 := [ 0 .. deduped_32_1 - 1 ];
+    deduped_24_1 := List( deduped_48_1, function ( i_2 )
+            return REM_INT( i_2, deduped_50_1 );
         end );
-    deduped_21_1 := SSortedList( List( [ 0 .. Length( deduped_22_1 ) - 1 ], function ( i_2 )
-              return deduped_20_1[1 + deduped_20_1[(1 + deduped_22_1[(1 + i_2)])]];
+    hoisted_23_1 := deduped_35_1{deduped_31_1};
+    hoisted_25_1 := List( deduped_29_1, function ( i_2 )
+              return deduped_24_1[1 + hoisted_23_1[(1 + i_2)]];
+          end ){deduped_31_1};
+    deduped_27_1 := SSortedList( List( deduped_29_1, function ( i_2 )
+              return deduped_24_1[1 + hoisted_25_1[(1 + i_2)]];
           end ) );
-    return CreateCapCategoryMorphismWithAttributes( cat_1, CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( Length( deduped_21_1 ) ) ), deduped_34_1, AsList, deduped_21_1 );
+    deduped_26_1 := BigInt( Length( deduped_27_1 ) );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, CreateCapCategoryObjectWithAttributes( cat_1, Length, deduped_26_1 ), deduped_53_1, AsList, deduped_27_1{[ 1 .. deduped_26_1 ]} );
 end
 ########
         
@@ -819,12 +870,16 @@ end
         
 ########
 function ( cat_1, arg2_1 )
-    if Length( arg2_1 ) = 0 then
-        return CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( 1 ) );
-    else
-        return arg2_1;
-    fi;
-    return;
+    local deduped_1_1;
+    deduped_1_1 := Length( arg2_1 );
+    return CreateCapCategoryObjectWithAttributes( cat_1, Length, IdFunc( function (  )
+                if deduped_1_1 = 0 then
+                    return BigInt( 1 );
+                else
+                    return deduped_1_1;
+                fi;
+                return;
+            end )(  ) );
 end
 ########
         
@@ -881,7 +936,7 @@ end
         
 ########
 function ( cat_1, T_1, P_1 )
-    return CreateCapCategoryMorphismWithAttributes( cat_1, P_1, T_1, AsList, [  ] );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, P_1, T_1, AsList, [ 0 .. Length( P_1 ) - 1 ] );
 end
 ########
         
@@ -894,18 +949,18 @@ end
 function ( cat_1, objects_1, T_1, tau_1, P_1 )
     local hoisted_1_1, hoisted_2_1, hoisted_3_1, deduped_5_1;
     deduped_5_1 := [ 0 .. Length( objects_1 ) - 1 ];
-    hoisted_2_1 := List( objects_1, Length );
-    hoisted_3_1 := List( deduped_5_1, function ( j_2 )
-            return Product( hoisted_2_1{[ 1 .. j_2 ]} );
+    hoisted_3_1 := List( tau_1, AsList );
+    hoisted_1_1 := List( objects_1, Length );
+    hoisted_2_1 := List( deduped_5_1, function ( j_2 )
+            return Product( hoisted_1_1{[ 1 .. j_2 ]} );
         end );
-    hoisted_1_1 := List( tau_1, AsList );
     return CreateCapCategoryMorphismWithAttributes( cat_1, T_1, P_1, AsList, List( [ 0 .. Length( T_1 ) - 1 ], function ( i_2 )
               local hoisted_1_2;
               hoisted_1_2 := 1 + i_2;
               return Sum( deduped_5_1, function ( j_3 )
                       local deduped_1_3;
                       deduped_1_3 := 1 + j_3;
-                      return hoisted_1_1[deduped_1_3][hoisted_1_2] * hoisted_3_1[deduped_1_3];
+                      return hoisted_2_1[deduped_1_3] * hoisted_3_1[deduped_1_3][hoisted_1_2];
                   end );
           end ) );
 end
@@ -966,7 +1021,7 @@ BindGlobal( "CategoryOfSkeletalFinSetsWithMorphismsGivenByListsPrecompiled", fun
         
         
         function (  )
-    return CategoryOfSkeletalFinSets(  );
+    return CategoryOfSkeletalFinSetsWithMorphismsGivenByLists(  );
 end;
         
         
diff --git a/init.g b/init.g
index 8cffd4b..52481bf 100644
--- a/init.g
+++ b/init.g
@@ -10,6 +10,8 @@ ReadPackage( "FinSetsForCAP", "gap/FinSets.gd" );
 
 ReadPackage( "FinSetsForCAP", "gap/SkeletalFinSets.gd" );
 
+ReadPackage( "FinSetsForCAP", "gap/SkeletalFinSetsEager.gd" );
+
 #= comment for Julia
 if IsPackageMarkedForLoading( "JuliaInterface", ">= 0.2" ) then
     # COVERAGE_IGNORE_NEXT_LINE
diff --git a/read.g b/read.g
index ee84fd6..dc35118 100644
--- a/read.g
+++ b/read.g
@@ -8,10 +8,14 @@ ReadPackage( "FinSetsForCAP", "gap/Tools.gi" );
 
 ReadPackage( "FinSetsForCAP", "gap/precompiled_categories/CategoryOfSkeletalFinSetsWithMorphismsGivenByListsPrecompiled.gi" );
 
+ReadPackage( "FinSetsForCAP", "gap/precompiled_categories/CategoryOfSkeletalFinSetsWithMorphismsGivenByFunctionsPrecompiled.gi" );
+
 ReadPackage( "FinSetsForCAP", "gap/FinSets.gi" );
 
 ReadPackage( "FinSetsForCAP", "gap/SkeletalFinSets.gi" );
 
+ReadPackage( "FinSetsForCAP", "gap/SkeletalFinSetsEager.gi" );
+
 #= comment for Julia
 if IsPackageMarkedForLoading( "JuliaInterface", ">= 0.2" ) then
     # COVERAGE_IGNORE_NEXT_LINE