  We show that, given a finitely generated group $G$ as the coordinate group of
a finite system of equations over a torsion-free hyperbolic group $\Gamma$,
there is an algorithm which constructs a cover of a canonical solution diagram.
The diagram encodes all homomorphisms from $G$ to $\Gamma$ as compositions of
factorizations through $\Gamma$-NTQ groups and canonical automorphisms of the
corresponding NTQ-subgroups. We also give another characterization of
$\Gamma$-limit groups as iterated generalized doubles over $\Gamma$.
