Amalgamated free product

Definition with strict common subgroup
Let $$G_1$$ and $$G_2$$ be two groups, and let $$H$$ be a group with an injective homomorphism to both $$G_1$$ and $$G_2$$. Then the amalgamated free product of $$G_1$$ and $$G_2$$ via $$H$$ is defined as the quotient of the free product of $$G_1$$ and $$G_2$$, by the relation of the $$H$$ in $$G_1$$ being the same as the $$H$$ in $$G_2$$.