  We prove that a homotopy cofinal functor between small categories induces a
weak equivalence between homotopy colimits of pointed simplicial sets. This is
used to prove that the non-Abelian homology of a group diagram is isomorphic to
the homology of its inverse image under a homotopy cofinal functor. This also
made it possible to establish that the non-Abelian homology of group diagrams
are invariant under the left Kan extension along virtual discrete cofibrations.
With the help of these results, we have constructed a non-Abelian
Gabriel-Zisman homology theory for simplicial sets with coefficients in group
diagrams. Moreover, we have generalized this theory to presheaves over an
arbitrary small category, which play the role of simplicial sets. Sufficient
conditions are found for the isomorphism of non-Abelian homology of presheaves
over a small category with coefficients in group diagrams. It is proved that
the non-Abelian homology of the factorization category for the small category
is isomorphic to the Baues-Wirsching homology. A condition is found under which
a functor between small categories induces an isomorphism of non-Abelian
Baues-Wirsching homology.
