Free Product with amalgamation

Let be groups presented as with
Define the free product of and as:

The universal property gives

Suppose also is a group and let and be group homomorphisms

Free product with amalgamation over :

Induced maps also called

Universal property of amalgamated free products

For any group , there is a bijection

1. from group homomorphisms
2. to group homomorphisms and st
   given by

Proof

Just expand the definitions … umm yea