Group homomorphism/Z to group/Fact

Let ${}G$ denote a group.

Then there is a correspondence between group elements

${}g\in G$ and group homomorphisms ${}\varphi$ from ${}\mathbb {Z}$ to ${}G$, given by

$g\longmapsto (n\mapsto g^{n}){\text{ and }}\varphi \longmapsto \varphi (1).$

Proof, Write another proof