diff --git a/CAP/PackageInfo.g b/CAP/PackageInfo.g
index 5e48a31270..cbdfb4f511 100644
--- a/CAP/PackageInfo.g
+++ b/CAP/PackageInfo.g
@@ -10,7 +10,7 @@ SetPackageInfo( rec(
 
 PackageName := "CAP",
 Subtitle := "Categories, Algorithms, Programming",
-Version := "2023.10-01",
+Version := "2023.10-02",
 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",
 
diff --git a/CAP/gap/DerivedMethods.autogen.gi b/CAP/gap/DerivedMethods.autogen.gi
index 3b9a2c5c49..2613f2696d 100644
--- a/CAP/gap/DerivedMethods.autogen.gi
+++ b/CAP/gap/DerivedMethods.autogen.gi
@@ -1048,6 +1048,21 @@ end : CategoryFilter := IsCapCategory,
       Weight := 1,
       is_autogenerated_by_CompilerForCAP := true );
 
+##
+AddDerivationToCAP( MorphismFromCoimageToImageWithGivenObjects,
+                    "dualizing the derivation of MorphismFromCoimageToImageWithGivenObjects by MorphismFromCoimageToImageWithGivenObjects using that the image embedding lifts the coimage astriction",
+                    [
+                        [ CoimageProjectionWithGivenCoimageObject, 1 ],
+                        [ CoastrictionToImageWithGivenImageObject, 1 ],
+                        [ ColiftAlongEpimorphism, 1 ],
+                    ],
+                    
+  function ( cat_1, C_1, alpha_1, I_1 )
+    return ColiftAlongEpimorphism( cat_1, CoimageProjectionWithGivenCoimageObject( cat_1, alpha_1, C_1 ), CoastrictionToImageWithGivenImageObject( cat_1, alpha_1, I_1 ) );
+end : CategoryFilter := IsCapCategory,
+      Weight := 1,
+      is_autogenerated_by_CompilerForCAP := true );
+
 ##
 AddDerivationToCAP( MorphismFromCoimageToImageWithGivenObjects,
                     "dualizing the derivation of MorphismFromCoimageToImageWithGivenObjects by MorphismFromCoimageToImageWithGivenObjects using that images are given by kernels of cokernels",
diff --git a/CAP/gap/DerivedMethods.gi b/CAP/gap/DerivedMethods.gi
index 4cfed3824e..be7ec909fe 100644
--- a/CAP/gap/DerivedMethods.gi
+++ b/CAP/gap/DerivedMethods.gi
@@ -2907,6 +2907,21 @@ AddFinalDerivationBundle( "IsomorphismFromImageObjectToKernelOfCokernel as the i
   end,
 ] : CategoryFilter := IsAbelianCategory );
 
+##
+AddDerivationToCAP( MorphismFromCoimageToImageWithGivenObjects,
+                    "MorphismFromCoimageToImageWithGivenObjects using that the image embedding lifts the coimage astriction",
+                    [ [ ImageEmbeddingWithGivenImageObject, 1 ],
+                      [ AstrictionToCoimageWithGivenCoimageObject, 1 ],
+                      [ LiftAlongMonomorphism, 1 ] ],
+                    
+  function( cat, coimage, morphism, image )
+    
+    return LiftAlongMonomorphism( cat,
+                   ImageEmbeddingWithGivenImageObject( cat, morphism, image ),
+                   AstrictionToCoimageWithGivenCoimageObject( cat, morphism, coimage ) );
+    
+end );
+
 ##
 AddDerivationToCAP( MorphismFromCoimageToImageWithGivenObjects,
                     "MorphismFromCoimageToImageWithGivenObjects using that images are given by kernels of cokernels",
diff --git a/LinearAlgebraForCAP/PackageInfo.g b/LinearAlgebraForCAP/PackageInfo.g
index 2207edf36b..95d8625240 100644
--- a/LinearAlgebraForCAP/PackageInfo.g
+++ b/LinearAlgebraForCAP/PackageInfo.g
@@ -10,7 +10,7 @@ SetPackageInfo( rec(
 
 PackageName := "LinearAlgebraForCAP",
 Subtitle := "Category of Matrices over a Field for CAP",
-Version := "2023.09-03",
+Version := "2023.10-01",
 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",
 
@@ -89,7 +89,7 @@ Dependencies := rec(
   NeededOtherPackages := [ [ "ToolsForHomalg", ">= 2015.09.18" ],
                            [ "MatricesForHomalg", ">= 2023.01-01" ],
                            [ "GaussForHomalg", ">= 2021.04-02" ],
-                           [ "CAP", ">= 2023.08-16" ],
+                           [ "CAP", ">= 2023.10-02" ],
                            [ "MonoidalCategories", ">= 2023.08-11" ],
                            ],
   SuggestedOtherPackages := [
diff --git a/LinearAlgebraForCAP/gap/precompiled_categories/MatrixCategoryPrecompiled.gi b/LinearAlgebraForCAP/gap/precompiled_categories/MatrixCategoryPrecompiled.gi
index 5a33bb1a17..52d35241be 100644
--- a/LinearAlgebraForCAP/gap/precompiled_categories/MatrixCategoryPrecompiled.gi
+++ b/LinearAlgebraForCAP/gap/precompiled_categories/MatrixCategoryPrecompiled.gi
@@ -2108,16 +2108,13 @@ end
         
 ########
 function ( cat_1, C_1, alpha_1, I_1 )
-    local morphism_attr_1_1, deduped_2_1, deduped_3_1, deduped_4_1;
-    deduped_4_1 := UnderlyingRing( cat_1 );
-    deduped_3_1 := UnderlyingMatrix( alpha_1 );
-    deduped_2_1 := SyzygiesOfRows( SyzygiesOfColumns( deduped_3_1 ) );
-    morphism_attr_1_1 := UniqueRightDivide( HomalgIdentityMatrix( RowRankOfMatrix( deduped_3_1 ), deduped_4_1 ), UniqueRightDivide( UniqueLeftDivide( BasisOfColumns( deduped_3_1 ), deduped_3_1 ), deduped_2_1 ) * UniqueRightDivide( HomalgIdentityMatrix( NumberRows( deduped_2_1 ), deduped_4_1 ), UniqueRightDivide( BasisOfRows( deduped_3_1 ), deduped_2_1 ) ) );
-    return CreateCapCategoryMorphismWithAttributes( cat_1, CreateCapCategoryObjectWithAttributes( cat_1, Dimension, NumberRows( morphism_attr_1_1 ) ), C_1, UnderlyingMatrix, morphism_attr_1_1 );
+    local deduped_1_1;
+    deduped_1_1 := UnderlyingMatrix( alpha_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, I_1, C_1, UnderlyingMatrix, UniqueRightDivide( HomalgIdentityMatrix( Dimension( I_1 ), UnderlyingRing( cat_1 ) ), UniqueRightDivide( UniqueLeftDivide( BasisOfColumns( deduped_1_1 ), deduped_1_1 ), BasisOfRows( deduped_1_1 ) ) ) );
 end
 ########
         
-    , 1610 : IsPrecompiledDerivation := true );
+    , 604 : IsPrecompiledDerivation := true );
     
     ##
     AddIsAutomorphism( cat,
@@ -4039,15 +4036,13 @@ end
         
 ########
 function ( cat_1, C_1, alpha_1, I_1 )
-    local morphism_attr_1_1, deduped_2_1, deduped_3_1;
-    deduped_3_1 := UnderlyingMatrix( alpha_1 );
-    deduped_2_1 := SyzygiesOfRows( SyzygiesOfColumns( deduped_3_1 ) );
-    morphism_attr_1_1 := UniqueRightDivide( UniqueLeftDivide( BasisOfColumns( deduped_3_1 ), deduped_3_1 ), deduped_2_1 ) * UniqueRightDivide( HomalgIdentityMatrix( NumberRows( deduped_2_1 ), UnderlyingRing( cat_1 ) ), UniqueRightDivide( BasisOfRows( deduped_3_1 ), deduped_2_1 ) );
-    return CreateCapCategoryMorphismWithAttributes( cat_1, C_1, CreateCapCategoryObjectWithAttributes( cat_1, Dimension, NumberColumns( morphism_attr_1_1 ) ), UnderlyingMatrix, morphism_attr_1_1 );
+    local deduped_1_1;
+    deduped_1_1 := UnderlyingMatrix( alpha_1 );
+    return CreateCapCategoryMorphismWithAttributes( cat_1, C_1, I_1, UnderlyingMatrix, UniqueRightDivide( UniqueLeftDivide( BasisOfColumns( deduped_1_1 ), deduped_1_1 ), BasisOfRows( deduped_1_1 ) ) );
 end
 ########
         
-    , 1408 : IsPrecompiledDerivation := true );
+    , 402 : IsPrecompiledDerivation := true );
     
     ##
     AddMorphismFromEqualizerToSink( cat,