Discrete opfibrations of graphs

Suppose you have coloured graphs G,H (with multiple edges, etc. i.e., we are in the topos of coloured graphs). A morphism G->H induces a functor between the free categories generated by G and H. I am interested in those morphisms which induce discrete opfibrations. Has anyone studied this notion? Essentially, any arc f(x)->y of H can be lifted uniquely to an arc (with the same label) x->x', for some x' such that f(x')=y. Sebastiano Vigna