Projection on Group Direct Product is Epimorphism/Proof 1

Proof
From Projection is Surjection, $\operatorname{pr}_1$ and $\operatorname{pr}_2$ are surjections.

We now need to show they are homomorphisms.

Let $g, h \in \left({G, \circ}\right)$ where $g = \left({g_1, g_2}\right)$ and $h = \left({h_1, h_2}\right)$.

Then:

and thus the morphism property is demonstrated for both $\operatorname{pr}_1$ and $\operatorname{pr}_2$.