diff --git a/doc/Doc.autodoc b/doc/Doc.autodoc
index 6f1cb1e..611e187 100644
--- a/doc/Doc.autodoc
+++ b/doc/Doc.autodoc
@@ -5,6 +5,9 @@
 @Subsection A category of module homomorphisms
 @InsertChunk CategoryOfRepresentations
 
+@Subsection Another category of module homomorphisms
+@InsertChunk RepresentingC4C4
+
 @Section GAP categories
 @Section Global variables
 @Section Attributes
diff --git a/examples/CategoryOfRepresentations.g b/examples/CategoryOfRepresentations.g
index 5e96ee7..cace767 100644
--- a/examples/CategoryOfRepresentations.g
+++ b/examples/CategoryOfRepresentations.g
@@ -159,6 +159,8 @@ Display( nine(kq.b) );
 #! A morphism in Category of matrices over GF(3)
 IsWellDefined( nine );
 #! true
+Length( WeakDirectSumDecomposition( nine ) );
+#! 1
 fortyone := TensorProductOnObjects( nine, nine );
 #! <(1)->25, (2)->16; (a)->25x25, (b)->25x16, (c)->16x16>
 IsWellDefined( fortyone );
diff --git a/examples/RepresentingC4C4.g b/examples/RepresentingC4C4.g
new file mode 100644
index 0000000..8224a71
--- /dev/null
+++ b/examples/RepresentingC4C4.g
@@ -0,0 +1,241 @@
+#! @BeginChunk RepresentingC4C4
+
+LoadPackage( "CatReps" );
+
+#! In order to understand the workings and choices in the first example,
+#! I made a similar example to compare the results.
+
+#! @Example
+c4c4 := ConcreteCategoryForCAP( [ [2,3,4,1], [5,6,7,8],
+ [,,,,6,7,8,5] ] );
+#! A finite concrete category
+HOMALG_MATRICES.PreferDenseMatrices := true;
+#! true
+GF3 := HomalgRingOfIntegers( 3 );
+#! GF(3)
+kq := Algebroid( GF3, c4c4 );
+#! Algebroid generated by the right quiver 
+#! q(2)[a:1->1,b:1->2,c:2->2]
+UnderlyingQuiverAlgebra( kq );
+#! (GF(3) * q) / 
+#! [ Z(3)^0*(a*a*a*a) + Z(3)*(1), 
+#!   Z(3)^0*(c*c*c*c) + Z(3)*(2),
+#!   Z(3)*(b*c) + Z(3)^0*(a*b) ]
+SetIsLinearClosureOfACategory( kq, true );
+#! @EndExample
+
+#! We are now looking to represent our concrete category
+#! $c4c4$, i.e. we want to find matrices a, b, c that
+#! satisfy the relations of the Algebroid.
+#! Since the endomorphisms are permutations on a set of
+#! 4 elements it makes sense to choose permutation matrices.
+#! But if we choose 4x4 permutation matrices for both
+#! endomorphisms, the only possible matrix for b would
+#! be the 4x4 identity matrix, which isn't very interesting.
+#! Instead we can see the permutations on 4 elements in 
+#! $S_4$ as elements of $Stab_{5,6}$ subset $S_6$ or of 
+#! $Stab_{5}$ subset $S_5$. As permutation matrices, they
+#! are block diagonal matrices with the smaller permutation
+#! matrix on 4 elements complemented with a 2x2 or 1x1
+#! identity matrix on the diagonal.
+#! This gives us two different
+#! dimensions for our target matrix category, $6$ and $5$
+#! respectively, and makes the choice of the matrix for b
+#! less trivial.
+
+#! @Example
+CatReps := Hom( kq, GF3 );
+#! The category of functors: Algebroid generated by the 
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> 
+#! Category of matrices over GF(3)
+#aMat := HomalgMatrix( [ 
+aMat :=[ 
+	[ 0,0,0,1,0,0 ], 
+	[ 1,0,0,0,0,0 ], 
+	[ 0,1,0,0,0,0 ], 
+	[ 0,0,1,0,0,0 ], 
+	[ 0,0,0,0,1,0 ], 
+	[ 0,0,0,0,0,1 ] 
+];;
+#	], 6, 6, GF3 );
+# <A 6 x 6 matrix over an internal ring>
+#cMat := HomalgMatrix( [
+cMat := [
+	[ 0,0,0,1,0 ], 
+	[ 1,0,0,0,0 ], 
+	[ 0,1,0,0,0 ], 
+	[ 0,0,1,0,0 ], 
+	[ 0,0,0,0,1 ] 
+];;
+#	], 5, 5, GF3 );
+# <A 5 x 5 matrix over an internal ring>
+Display( aMat^4 );
+#! [ [  1,  0,  0,  0,  0,  0 ],
+#!  [  0,  1,  0,  0,  0,  0 ],
+#!  [  0,  0,  1,  0,  0,  0 ],
+#!  [  0,  0,  0,  1,  0,  0 ],
+#!  [  0,  0,  0,  0,  1,  0 ],
+#!  [  0,  0,  0,  0,  0,  1 ] ]
+Display( cMat^4 );
+#! [ [  1,  0,  0,  0,  0 ],
+#!  [  0,  1,  0,  0,  0 ],
+#!  [  0,  0,  1,  0,  0 ],
+#!  [  0,  0,  0,  1,  0 ],
+#!  [  0,  0,  0,  0,  1 ] ]
+#! @EndExample
+
+#! I calculated by hand the matrix multiplications
+#! $a*b = b*c$ and what they imply for the entries in b
+#! given the matrices a and c above. The result is that
+#! from the 30 variables $b_{11},...,b_{65}$ only
+#! nine free variables remain, which can be set to
+#! any element in our field:
+#!
+
+#! @Example
+b11 := 0;; b12 := 1;; b13 := 0;; b21 := 1;;
+b15 := 0;; b51 := 1;; b55 := 0;; b61 := 1;; b65 := 1;;
+
+#bMat := HomalgMatrix( [ 
+bMat := [
+	[b11,b12,b13,b21,b15], 
+	[b21,b11,b12,b13,b15], 
+	[b13,b21,b11,b12,b15], 
+	[b12,b13,b21,b11,b15], 
+	[b51,b51,b51,b51,b55], 
+	[b61,b61,b61,b61,b65] 
+];;
+#], 6,5, GF3 );
+# <A 6 x 5 matrix over an internal ring>
+Display( bMat );
+#! [ [  0,  1,  0,  1,  0 ],
+#!  [  1,  0,  1,  0,  0 ],
+#!  [  0,  1,  0,  1,  0 ],
+#!  [  1,  0,  1,  0,  0 ],
+#!  [  1,  1,  1,  1,  0 ],
+#!  [  1,  1,  1,  1,  1 ] ]
+Display( aMat*bMat );
+#! [ [  1,  0,  1,  0,  0 ],
+#! [  0,  1,  0,  1,  0 ],
+#! [  1,  0,  1,  0,  0 ],
+#! [  0,  1,  0,  1,  0 ],
+#! [  1,  1,  1,  1,  0 ],
+#! [  1,  1,  1,  1,  1 ] ]
+Display( bMat*cMat );
+#! [ [  1,  0,  1,  0,  0 ],
+#! [  0,  1,  0,  1,  0 ],
+#! [  1,  0,  1,  0,  0 ],
+#! [  0,  1,  0,  1,  0 ],
+#! [  1,  1,  1,  1,  0 ],
+#! [  1,  1,  1,  1,  1 ] ]
+aMat*bMat = bMat*cMat;
+#! true
+#! @EndExample
+
+#! So the three relations in our algebroid should be satisfied
+#! by these matrices, therefore they should make a
+#! well-defined representation of C4C4. (Defining the matrices
+#! as HomalgMatrices instead of GAP matrices yields in an error
+#! in RecordOfCatRep which is used in WeakDirectSumDecomposition,
+#! so for now they are defined as GAP matrices).
+
+#! @Example
+eleven := AsObjectInHomCategory( kq, [ 6, 5 ], [ aMat, bMat, cMat ] );
+#! <(1)->6, (2)->5; (a)->6x6, (b)->6x5, (c)->5x5>
+IsWellDefined( eleven );
+#! true
+Display( eleven );
+#! An object in The category of functors: Algebroid
+#! generated by the right quiver q(2)[a:1->1,b:1->2,c:2->2]
+#! -> Category of matrices over GF(3) defined by the following data:
+#!
+#!
+#! Image of <(1)>:
+#! A vector space object over GF(3) of dimension 6
+#!
+#! Image of <(2)>:
+#! A vector space object over GF(3) of dimension 5
+#!
+#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
+#! . . . 1 . .
+#! 1 . . . . .
+#! . 1 . . . .
+#! . . 1 . . .
+#! . . . . 1 .
+#! . . . . . 1
+#!
+#! A morphism in Category of matrices over GF(3)
+#!
+#!
+#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
+#! . 1 . 1 .
+#! 1 . 1 . .
+#! . 1 . 1 .
+#! 1 . 1 . .
+#! 1 1 1 1 .
+#! 1 1 1 1 1
+#!
+#! A morphism in Category of matrices over GF(3)
+#!
+#!
+#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
+#! . . . 1 .
+#! 1 . . . .
+#! . 1 . . .
+#! . . 1 . .
+#! . . . . 1
+#!
+#! A morphism in Category of matrices over GF(3)
+gammas := WeakDirectSumDecomposition( eleven );
+#! [ <(1)->1x6, (2)->0x5>, <(1)->1x6, (2)->1x5>, <(1)->1x6, (2)->1x5>,
+#!  <(1)->2x6, (2)->0x5>, <(1)->0x6, (2)->2x5>, <(1)->1x6, (2)->1x5> ]
+#! @EndExample
+
+#! As opposed to nine in the first example, eleven itself
+#! can already be decomposed.
+
+#! @Example
+Display( Source( UniversalMorphismFromDirectSum( gammas ) ) );
+#! An object in The category of functors: Algebroid 
+#! generated by the right quiver q(2)[a:1->1,b:1->2,c:2->2]
+#! -> Category of matrices over GF(3) defined by the following data:
+#!
+#!
+#! Image of <(1)>:
+#! A vector space object over GF(3) of dimension 6
+#!
+#! Image of <(2)>:
+#! A vector space object over GF(3) of dimension 5
+#!
+#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
+#! 1 . . . . .
+#! . 1 . . . .
+#! . . 2 . . .
+#! . . . . 2 .
+#! . . . 1 . .
+#! . . . . . 1
+#!
+#! A morphism in Category of matrices over GF(3)
+#!
+#!
+#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
+#! . . . . .
+#! 1 . . . .
+#! . 1 . . .
+#! . . . . .
+#! . . . . .
+#! . . . . 2
+#!
+#! A morphism in Category of matrices over GF(3)
+#!
+#!
+#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
+#! 1 . . . .
+#! . 2 . . .
+#! . . 2 2 .
+#! . . 2 1 .
+#! . . . . 1
+#!
+#! A morphism in Category of matrices over GF(3)
+#! @EndExample
+#! @EndChunk