Projection is Epimorphism

Theorem
Let $\left({S, \circ}\right)$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right)$ and $\left({S_2, \circ_2}\right)$.

Then:
 * $\operatorname{pr}_1$ is an epimorphism from $\left({S, \circ}\right)$ to $\left({S_1, \circ_1}\right)$
 * $\operatorname{pr}_2$ is an epimorphism from $\left({S, \circ}\right)$ to $\left({S_2, \circ_2}\right)$

where $\operatorname{pr}_1$ and $\operatorname{pr}_2$ are the first and second projection respectively of $\left({S, \circ}\right)$.

Proof
From Projections are Surjections, $\operatorname{pr}_1$ and $\operatorname{pr}_2$ are surjections.

We now need to show they are homomorphisms.

Let $s, t \in \left({S, \circ}\right)$ where $s = \left({s_1, s_2}\right)$ and $t = \left({t_1, t_2}\right)$.

Then:

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