Universal property of a pushout of groups ($\mathrm{Hom}(G_1{\ast }_{G_0}{G}_2,H)$)

For group homomorphisms ${\varphi }_j:G_0\to G_j$, a group homomorphism from their pushout of groups to $H$ is uniquely a pair $u_j:G_j\to H$ with $u_1{\varphi }_1=u_2{\varphi }_2$. Indeed extend the pair to the free product and factor through its quotient group: the agreement kills the normal closure of all ${\varphi }_1(g){\varphi }_2(g{)}^{-1}$. Uniqueness follows because the images of the two factors generate the pushout of groups.