According to the definition:

A graph (*V*, *A*, *src*, *tgt*) involves two sets and two functions. For two graphs to be comparable, their two sets and their two functions should be appropriately comparable. Let *G* = (*V*, *A*, *src*, *tgt*) and *G*′ = (*V*′, *A*′, *src*′, *tgt*′) be graphs. A *graph homomorphism f from G to G*′, denoted *f* : *G* → *G*′, consists of two functions *f*_{0}: *V* → *V*′ and *f*_{1}: *A* → *A*′ …the rest…

So this means that not all elements of A’ need to have a mapping from A and not all elements of V’ need to have a mapping from V. For example if G has m vertices and n edges and G’ has m'(>1) vertices and n'(>1) edges we could have all vertices of G go to one vertex of G'(say a) and all edges of G go to an edge from a to a(assuming that exists) in G’. That would be a valid homomorphism. Am I understanding that correctly ?