Group Homomorphism Preserves Identity

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 1
The result follows directly from the morphism property of $$\circ$$ under $$\phi$$:

$$ $$

That is:

$$ $$

Proof 2
A direct application of Homomorphism to Group Preserves Identity and Inverses.