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 homomorphism.

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

Proof
The result for $$n \in \mathbb{N}^*$$ 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.