Suppose we are given groups $G_1,G_2$ and some subgroup $H$ of $G_1$ and $G_2$ and let $G_1*_HG_2$ denote the amalgamated free product.
I stumbled upon the following claim:
$G$ is generated by $G_1\sqcup G_2$ $\iff$ $G_1*_HG_2 \to G$ is surjective.
Now i've tried to think about this in terms of the universal property together with the definition of the amalgamated free product as the quotient of a free group $F\langle G_1,G_2\rangle$ modulo the usual relations. But i still struggle to see why this is true.
Does this follow from descending to the quotient? Could someone elaborate?
Side-Question: Is $F\langle G_1,G_2\rangle$ the same as $F\langle G_1\sqcup G_2\rangle$?
("$\sqcup$" denotes the disjoint union)