diff --git a/CAP/gap/DerivedMethods.gi b/CAP/gap/DerivedMethods.gi
index a6f8362ffe..9b91480722 100644
--- a/CAP/gap/DerivedMethods.gi
+++ b/CAP/gap/DerivedMethods.gi
@@ -1820,6 +1820,26 @@ AddDerivationToCAP( IsomorphismFromKernelOfCokernelToImageObject,
     
 end : Description := "IsomorphismFromKernelOfCokernelToImageObject as the inverse of IsomorphismFromImageObjectToKernelOfCokernel" );
 
+##
+AddDerivationToCAP( IsomorphismFromKernelOfCokernelToImageObject,
+                    [ [ ImageEmbedding, 1 ],
+                      [ CokernelProjection, 1 ],
+                      [ KernelEmbedding, 1 ],
+                      [ LiftAlongMonomorphism, 1 ] ],
+                      
+  function( cat, mor )
+    local image_embedding, ker_of_coker_embedding;
+    
+    image_embedding := ImageEmbedding( cat, mor );
+    
+    ker_of_coker_embedding := KernelEmbedding( cat, CokernelProjection( cat, mor ) );
+    
+    return LiftAlongMonomorphism( cat, image_embedding, ker_of_coker_embedding );
+    
+  end : CategoryFilter := IsAbelianCategory, ##FIXME: PreAbelian?
+      Description := "IsomorphismFromKernelOfCokernelToImageObject as the unique lift of the kernel of the cokernel along the image embedding"
+);
+
 ##
 AddDerivationToCAP( IsomorphismFromImageObjectToKernelOfCokernel,
         
@@ -1875,6 +1895,26 @@ AddDerivationToCAP( IsomorphismFromCoimageToCokernelOfKernel,
     
 end : Description := "IsomorphismFromCoimageToCokernelOfKernel as the inverse of IsomorphismFromCokernelOfKernelToCoimage" );
 
+##
+AddDerivationToCAP( IsomorphismFromCoimageToCokernelOfKernel,
+                    [ [ CoimageProjection, 1 ],
+                      [ KernelEmbedding, 1 ],
+                      [ CokernelProjection, 1 ],
+                      [ ColiftAlongEpimorphism, 1 ] ],
+                      
+  function( cat, mor )
+    local coimage_projection, coker_of_ker_projection;
+    
+    coimage_projection := CoimageProjection( cat, mor );
+    
+    coker_of_ker_projection := CokernelProjection( cat, KernelEmbedding( cat, mor ) );
+    
+    return ColiftAlongEpimorphism( cat, coimage_projection, coker_of_ker_projection );
+    
+  end : CategoryFilter := IsAbelianCategory, ##FIXME: PreAbelian?
+      Description := "IsomorphismFromCoimageToCokernelOfKernel as the unique colift of the cokernel of the kernel along the coimage projection"
+);
+
 ##
 AddDerivationToCAP( IsomorphismFromFiberProductToKernelOfDiagonalDifference,
           
diff --git a/LinearAlgebraForCAP/gap/CompilerLogic.gi b/LinearAlgebraForCAP/gap/CompilerLogic.gi
index 5e0e1a81cd..ef22dc5e71 100644
--- a/LinearAlgebraForCAP/gap/CompilerLogic.gi
+++ b/LinearAlgebraForCAP/gap/CompilerLogic.gi
@@ -54,3 +54,13 @@ CapJitAddLogicTemplate(
         needed_packages := [ [ "MatricesForHomalg", ">= 2020.05.19" ] ],
     )
 );
+
+# RightDivide( B, A ) * RightDivide( A, C ) => RightDivide( B, C )
+CapJitAddLogicTemplate(
+    rec(
+        variable_names := [ "A", "B", "C" ],
+        variable_filters := [ "IsHomalgMatrix", "IsHomalgMatrix", "IsHomalgMatrix" ],
+        src_template := "RightDivide( B, A ) * RightDivide( A, C )",
+        dst_template := "RightDivide( B, C )",
+    )
+);
diff --git a/LinearAlgebraForCAP/gap/precompiled_categories/MatrixCategoryPrecompiled.gi b/LinearAlgebraForCAP/gap/precompiled_categories/MatrixCategoryPrecompiled.gi
index daaf2be544..1577c07291 100644
--- a/LinearAlgebraForCAP/gap/precompiled_categories/MatrixCategoryPrecompiled.gi
+++ b/LinearAlgebraForCAP/gap/precompiled_categories/MatrixCategoryPrecompiled.gi
@@ -2165,16 +2165,14 @@ 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 := RightDivide( HomalgIdentityMatrix( RowRankOfMatrix( deduped_3_1 ), deduped_4_1 ), RightDivide( LeftDivide( BasisOfColumns( deduped_3_1 ), deduped_3_1 ), deduped_2_1 ) * RightDivide( HomalgIdentityMatrix( NumberRows( deduped_2_1 ), deduped_4_1 ), RightDivide( BasisOfRows( deduped_3_1 ), deduped_2_1 ) ) );
+    local morphism_attr_1_1, deduped_2_1;
+    deduped_2_1 := UnderlyingMatrix( alpha_1 );
+    morphism_attr_1_1 := RightDivide( HomalgIdentityMatrix( RowRankOfMatrix( deduped_2_1 ), UnderlyingRing( cat_1 ) ), RightDivide( LeftDivide( BasisOfColumns( deduped_2_1 ), deduped_2_1 ), BasisOfRows( deduped_2_1 ) ) );
     return CreateCapCategoryMorphismWithAttributes( cat_1, CreateCapCategoryObjectWithAttributes( cat_1, Dimension, NumberRows( morphism_attr_1_1 ) ), C_1, UnderlyingMatrix, morphism_attr_1_1 );
 end
 ########
         
-    , 1817 : IsPrecompiledDerivation := true );
+    , 1512 : IsPrecompiledDerivation := true );
     
     ##
     AddIsAutomorphism( cat,
@@ -2603,12 +2601,12 @@ end
 function ( cat_1, alpha_1 )
     local morphism_attr_1_1, deduped_2_1;
     deduped_2_1 := UnderlyingMatrix( alpha_1 );
-    morphism_attr_1_1 := RightDivide( HomalgIdentityMatrix( ColumnRankOfMatrix( deduped_2_1 ), UnderlyingRing( cat_1 ) ), LeftDivide( SyzygiesOfColumns( SyzygiesOfRows( deduped_2_1 ) ), BasisOfColumns( deduped_2_1 ) ) );
+    morphism_attr_1_1 := LeftDivide( BasisOfColumns( deduped_2_1 ), SyzygiesOfColumns( SyzygiesOfRows( deduped_2_1 ) ) );
     return CreateCapCategoryMorphismWithAttributes( cat_1, CreateCapCategoryObjectWithAttributes( cat_1, Dimension, NumberRows( morphism_attr_1_1 ) ), CreateCapCategoryObjectWithAttributes( cat_1, Dimension, NumberColumns( morphism_attr_1_1 ) ), UnderlyingMatrix, morphism_attr_1_1 );
 end
 ########
         
-    , 808 : IsPrecompiledDerivation := true );
+    , 503 : IsPrecompiledDerivation := true );
     
     ##
     AddIsomorphismFromCokernelOfDiagonalDifferenceToPushout( cat,
@@ -2984,15 +2982,14 @@ end
         
 ########
 function ( cat_1, alpha_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 := RightDivide( HomalgIdentityMatrix( NumberRows( deduped_2_1 ), UnderlyingRing( cat_1 ) ), RightDivide( BasisOfRows( deduped_3_1 ), deduped_2_1 ) );
+    local morphism_attr_1_1, deduped_2_1;
+    deduped_2_1 := UnderlyingMatrix( alpha_1 );
+    morphism_attr_1_1 := RightDivide( SyzygiesOfRows( SyzygiesOfColumns( deduped_2_1 ) ), BasisOfRows( deduped_2_1 ) );
     return CreateCapCategoryMorphismWithAttributes( cat_1, CreateCapCategoryObjectWithAttributes( cat_1, Dimension, NumberRows( morphism_attr_1_1 ) ), CreateCapCategoryObjectWithAttributes( cat_1, Dimension, NumberColumns( morphism_attr_1_1 ) ), UnderlyingMatrix, morphism_attr_1_1 );
 end
 ########
         
-    , 808 : IsPrecompiledDerivation := true );
+    , 503 : IsPrecompiledDerivation := true );
     
     ##
     AddIsomorphismFromKernelOfDiagonalDifferenceToFiberProduct( cat,
@@ -4139,15 +4136,14 @@ 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 := RightDivide( LeftDivide( BasisOfColumns( deduped_3_1 ), deduped_3_1 ), deduped_2_1 ) * RightDivide( HomalgIdentityMatrix( NumberRows( deduped_2_1 ), UnderlyingRing( cat_1 ) ), RightDivide( BasisOfRows( deduped_3_1 ), deduped_2_1 ) );
+    local morphism_attr_1_1, deduped_2_1;
+    deduped_2_1 := UnderlyingMatrix( alpha_1 );
+    morphism_attr_1_1 := RightDivide( LeftDivide( BasisOfColumns( deduped_2_1 ), deduped_2_1 ), BasisOfRows( deduped_2_1 ) );
     return CreateCapCategoryMorphismWithAttributes( cat_1, C_1, CreateCapCategoryObjectWithAttributes( cat_1, Dimension, NumberColumns( morphism_attr_1_1 ) ), UnderlyingMatrix, morphism_attr_1_1 );
 end
 ########
         
-    , 1614 : IsPrecompiledDerivation := true );
+    , 1309 : IsPrecompiledDerivation := true );
     
     ##
     AddMorphismFromFiberProductToSink( cat,