Homomorphism of Power of Group Element

Theorem
Let $\left({G, \circ}\right)$ and $\left({H, \ast}\right)$ be groups.

Let $\phi: S \to T$ be a group homomorphism.

Then $\forall n \in \Z: \forall g \in G: \phi \left({g^n}\right) = \left({\phi \left({g}\right)}\right)^n$.

Proof
The result for $n \in \N^*$ follows directly from General Morphism Property for Semigroups.

For $n = 0$, we use Homomorphism with Cancellable Range Preserves Identity.

For $n < 0$, we use Homomorphism with Identity Preserves Inverses, along with Index Laws for Monoids: Negative Index.