diff --git a/PackageInfo.g b/PackageInfo.g
index 9f70ee55..b8562964 100644
--- a/PackageInfo.g
+++ b/PackageInfo.g
@@ -10,7 +10,7 @@ SetPackageInfo( rec(
 
 PackageName := "FinSetsForCAP",
 Subtitle := "The elementary topos of (skeletal) finite sets",
-Version := "2023.10-02",
+Version := "2023.10-03",
 
 Date := (function ( ) if IsBound( GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE ) then return GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE; else return Concatenation( ~.Version{[ 1 .. 4 ]}, "-", ~.Version{[ 6, 7 ]}, "-01" ); fi; end)( ),
 License := "GPL-2.0-or-later",
@@ -100,7 +100,7 @@ Dependencies := rec(
   NeededOtherPackages := [
                    [ "CAP", ">= 2023.09-11" ],
                    [ "CartesianCategories", ">= 2023.08-13" ],
-                   [ "Toposes", ">= 2023.10-01" ],
+                   [ "Toposes", ">= 2023.10-13" ],
                    ],
   SuggestedOtherPackages := [ ],
   ExternalConditions := [ ],
diff --git a/doc/Doc.autodoc b/doc/Doc.autodoc
index ecd98b6a..b72ebf13 100644
--- a/doc/Doc.autodoc
+++ b/doc/Doc.autodoc
@@ -106,6 +106,9 @@
 @InsertChunk SkeletalImageEmbedding
 @InsertChunk SkeletalCoastrictionToImage
 
+@Subsection Skeletal Coimage
+@InsertChunk SkeletalCoimageObject
+
 @Subsection Skeletal Preimage
 @InsertChunk SkeletalPreimage
 
diff --git a/examples/CoimageObject.g b/examples/CoimageObject.g
index c71a131c..6768915c 100644
--- a/examples/CoimageObject.g
+++ b/examples/CoimageObject.g
@@ -22,10 +22,11 @@ IsEpimorphism( phi );
 #! false
 IsSplitEpimorphism( phi );
 #! false
-iota := AstrictionToCoimage( phi );
-#! <A morphism in FinSets>
 pi := CoimageProjection( phi );
 #! <An epimorphism in FinSets>
+iota := AstrictionToCoimage( phi );
+#! <A morphism in FinSets>
 PreCompose( pi, iota ) = phi;
 #! true
+
 #! @EndExample
diff --git a/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByLists.g b/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByLists.g
index dd0ac451..9076c5a7 100644
--- a/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByLists.g
+++ b/examples/PrecompileCategoryOfSkeletalFinSetsWithMorphismsGivenByLists.g
@@ -8,7 +8,7 @@
 
 LoadPackage( "FinSetsForCAP", false );
 #! true
-LoadPackage( "CompilerForCAP", false );
+LoadPackage( "CompilerForCAP", ">= 2023.10-03", false );
 #! true
 
 ReadPackage( "FinSetsForCAP", "gap/CompilerLogic.gi" );
@@ -25,6 +25,7 @@ primitive_operations :=
 list_of_operations :=
   SortedList( Concatenation( primitive_operations, [
           "CoastrictionToImage",
+          "CoimageProjection",
           "IsHomSetInhabited",
           "TruthMorphismOfImplies",
           #"HasPushoutComplement",
diff --git a/examples/SkeletalCoimageObject.g b/examples/SkeletalCoimageObject.g
new file mode 100644
index 00000000..b1a2b1d4
--- /dev/null
+++ b/examples/SkeletalCoimageObject.g
@@ -0,0 +1,32 @@
+#! @Chunk SkeletalCoimageObject
+
+#! @Example
+LoadPackage( "FinSetsForCAP", false );
+#! true
+m := FinSet( 3 );
+#! |3|
+n := FinSet( 3 );
+#! |3|
+phi := MapOfFinSets( m, [ 1, 0, 1 ], n );
+#! |3| → |3|
+I := CoimageObject( phi );
+#! |2|
+IsMonomorphism( phi );
+#! false
+IsSplitMonomorphism( phi );
+#! false
+IsEpimorphism( phi );
+#! false
+IsSplitEpimorphism( phi );
+#! false
+pi := CoimageProjection( phi );
+#! |3| → |2|
+iota := AstrictionToCoimage( phi );
+#! |2| → |3|
+PreCompose( pi, iota ) = phi;
+#! true
+Display( iota );
+#! { 0, 1 } ⱶ[ 1, 0 ]→ { 0, 1, 2 }
+Display( ImageEmbedding( phi ) );
+#! { 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1, 2 }
+#! @EndExample
diff --git a/gap/SkeletalFinSets.gi b/gap/SkeletalFinSets.gi
index 5904f1b9..481de57f 100644
--- a/gap/SkeletalFinSets.gi
+++ b/gap/SkeletalFinSets.gi
@@ -486,6 +486,22 @@ AddImageEmbedding( SkeletalFinSets,
     
 end );
 
+##
+AddAstrictionToCoimage( SkeletalFinSets,
+  function ( cat, phi )
+    local L, map;
+    
+    L := AsList( phi );
+    
+    ## unlike ImageObject which is a subobject of the range,
+    ## the CoimageObject is a factor object of the source,
+    ## and we want to retain the sorting of the source:
+    map := DuplicateFreeList( L );
+    
+    return MorphismConstructor( cat, ObjectConstructor( cat, BigInt( Length( map ) ) ), map, Range( phi ) );
+    
+end );
+
 ## Limits
 
 ##
diff --git a/gap/precompiled_categories/SkeletalCategoryOfFiniteSetsWithMorphismsGivenByListsPrecompiled.gi b/gap/precompiled_categories/SkeletalCategoryOfFiniteSetsWithMorphismsGivenByListsPrecompiled.gi
index 32f6d214..1c36790d 100644
--- a/gap/precompiled_categories/SkeletalCategoryOfFiniteSetsWithMorphismsGivenByListsPrecompiled.gi
+++ b/gap/precompiled_categories/SkeletalCategoryOfFiniteSetsWithMorphismsGivenByListsPrecompiled.gi
@@ -5,6 +5,19 @@
 #
 BindGlobal( "ADD_FUNCTIONS_FOR_SkeletalCategoryOfFiniteSetsWithMorphismsGivenByListsPrecompiled", function ( cat )
     
+    ##
+    AddAstrictionToCoimage( cat,
+        
+########
+function ( cat_1, alpha_1 )
+    local deduped_1_1;
+    deduped_1_1 := DuplicateFreeList( AsList( alpha_1 ) );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( Length( deduped_1_1 ) ) ), Range( alpha_1 ), AsList, deduped_1_1 );
+end
+########
+        
+    , 100 );
+    
     ##
     AddCartesianBraidingInverseWithGivenDirectProducts( cat,
         
@@ -184,6 +197,23 @@ end
         
     , 100 );
     
+    ##
+    AddCoimageProjection( cat,
+        
+########
+function ( cat_1, alpha_1 )
+    local deduped_3_1, deduped_4_1, deduped_5_1;
+    deduped_5_1 := AsList( alpha_1 );
+    deduped_4_1 := Source( alpha_1 );
+    deduped_3_1 := DuplicateFreeList( deduped_5_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, deduped_4_1, CreateCapCategoryObjectWithAttributes( cat_1, Length, BigInt( Length( deduped_3_1 ) ) ), AsList, List( [ 0 .. Length( deduped_4_1 ) - 1 ], function ( x_2 )
+              return -1 + BigInt( SafePosition( deduped_3_1, deduped_5_1[(1 + x_2)] ) );
+          end ) );
+end
+########
+        
+    , 202 : IsPrecompiledDerivation := true );
+    
     ##
     AddColift( cat,