Let $G_1$, $G_2$ and $H$ be groups and $\phi_1: H \to G_1$ and $\phi_2: H \to G_2$ two monomorphisms. Consider the group $(G_1 \times G_2)/N$ where $N$ is the smallest normal subgroup containing the set $\{\big( \phi_1(h),\phi_2(h)\big)~|~h \in H\}$ $($here $G_1 \times G_2$ is direct product of groups$)$.
I want to know does this kind of "product" has some standard name. Can someone give some references? I want to know about their "properties".
I know I am not precise but any kind of help will be appreciated. Moreover, if someone has better title for this question please do the needful.