Amalgamated free product
= Amalgamated free product
Given homomorphisms $i_A:C\to A$ and $i_B:C\to B$, the amalgamated free product $A*_CB$ comes with homomorphisms from $A$ and $B$ that agree on $C$. It is universal with this property: every compatible pair of homomorphisms $A\to H$ and $B\to H$ factors uniquely through $A*_CB$.