Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be graphs and let $h:V_2\to V_1$ be a map such that, for every edge $(u_1,v_1)\in E_1$, there is an edge $(u_2,v_2)\in E_2$ such that $h(u_2)=u_1$ and $h(v_2)=v_1$.
This kind of maps has occurred in our recent research in Database Theory. We think that it looks like a natural "dual" to graph homomorphisms, where $h$ would be $V_1\to V_2$ and map each edge of $G_1$ to an edge of $G_2$. However, we could not find it in the literature, but, since we do not have a name for it, maybe we searched in the wrong places.
Some background: in Database Theory, homomorphisms play an important role in the context of the containment problem for conjunctive queries. The "kind-of-dual" form occurs in the context of parallel query evaluation.
In this slightly extended context, testing for existence of such a mapping is NP-complete, and I expect that also for graphs. However, I am not asking for complexity results, but I would be be grateful for pointers to the literature (or even a name, for that matter).