Group Homomorphisms Preserve the Identity Element and Inverses

Let $(G,\ast_G)$ and $(H,\ast_H)$ be groups with identity elements $e_G$ and $e_H$ and inverses as in Uniqueness of the Identity Element and of Inverses in a Group, and let $\varphi:G\to H$ be a group homomorphism. Then

$$\varphi(e_G)=e_H,$$

and for every $a\in G$

$$\varphi(a^{-1})=\varphi(a)^{-1},$$

where on the left the inverse is taken in $G$ and on the right in $H$.