Group Homomorphism Preserves Identity/Proof 1

Theorem
Let $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ be a group homomorphism.

Let:
 * $e_G$ be the identity of $G$
 * $e_H$ be the identity of $H$.

Then $\phi \left({e_G}\right) = e_H$.

Proof
The result follows directly from the morphism property of $\circ$ under $\phi$:

That is: