How does the amalgamated product not depend on the homomorphisms $\alpha$ and $\beta$?

Question

I came across this definition of the amalgamated product in my Algebraic Topology class:

I was curious as to why the choice of the homomorphisms $\alpha$ and $\beta$ is unimportant to the outcome of the group. Wouldn't the images $i_{G}(\alpha(K))$ and $i_H(\beta(K))$ change depending on these homomorphisms, resulting in different relations for the generators of $G*_KH$?

Answer 1

It does depend on $\alpha$ and $\beta$. It's just customary to leave them out of the notation $G*_KH$, since otherwise it would become a bit unwieldy.