Pushout in a category ($B{⨿}_{A}C$)

A pushout of $B\leftarrow A\to C$ is a colimit with morphisms $B\to P$ and $C\to P$ making the square commute and universal among such commuting pairs. In the Category of sets, form the disjoint union of $B,C$ and identify the two images of each element of $A$.

For homomorphisms $i_j:H\to G_j$, the pushout of groups is $(G_1\ast G_2)/⟨⟨i_1(h)i_2(h{)}^{-1}:h\in H⟩⟩$. It has the universal property for homomorphisms out of $G_1,G_2$ agreeing on $H$. When the maps $i_j$ are injective, it is the amalgamated free product, with the factors and common subgroup embedded.

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.