free product with amalgamated subgroup

Definition 1.

Let $G_{k}$ , $k=0,1,2$ be groups and $i_{k}\\colon\\thinspace G_{0}\\to G_{i}$ , $k=1,2$ 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 followingtwo properties

1.
there are homomorphisms $j_{k}\\colon\\thinspace G_{k}\\to G$ , $k=1,2$ that make the followingdiagram commute

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