free product with amalgamated subgroup
Definition 1.
Let ${G}_{k}$, $k=0,1,2$ be groups and ${i}_{k}:{G}_{0}\to {G}_{i}$, $k=1,2$ be monomorphisms^{}. The free product^{} of ${G}_{1}$ and ${G}_{2}$ with amalgamated subgroup^{} ${G}_{0}$, is defined to be a group $G$ that has the following two properties

1.
there are homomorphisms^{} ${j}_{k}:{G}_{k}\to G$, $k=1,2$ that make the following diagram commute