Effective embedding of residually hyperbolic groups into direct products of extensions of centralizers

For any torsion-free hyperbolic group Gamma and any group G that is fully residually Gamma, we construct algorithmically a finite collection of homomorphisms from G to groups obtained from Gamma by extensions of centralizers, at least one of which is injective. When G is residually Gamma, this gives a effective embedding of G into a direct product of such groups. We also give an algorithmic construction of a diagram encoding the set of homomorphisms from a given finitely presented group to Gamma.