= Universal property of a pushout of groups
{title2=$\operatorname{Hom}(G_1*_{G_0}G_2,H)$}
For <group homomorphisms> $\phi_j:G_0\to G_j$, a <group homomorphism> from their <pushout of groups> to $H$ is uniquely a pair $u_j:G_j\to H$ with $u_1\phi_1=u_2\phi_2$. Indeed extend the pair to the <free product> and factor through its <quotient group>: the agreement kills the <normal closure> of all $\phi_1(g)\phi_2(g)^{-1}$. Uniqueness follows because the images of the two factors generate the <pushout of groups>.
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