Let $G_1$ and $G_2$ be two abelian groups with respective subgroups $N_1$ and $N_2$. and let $f:G_1 \to G_2$ be a homomorphism. Under what conditions is the induced map $f':G_1/N_1 \to G_2/N_2$ well defined?
I think the only thing to verify is that given any $g_1\in N_1$ we must have $f(g_1)\in N_2$. is there any other condition?