Group homomorphism/Inverse to inverse/Fact

Let $G$ and $H$ denote groups, and let $\phi :G\to H$ be a group homomorphism.

Then

$\phi (e_G)=e_H$ and $(\phi (g){)}^{-1}=\phi \left(g^{-1}\right)$ for every

$g\in G$.

Proof, Write another proof