Identity Function is Completely Multiplicative

Theorem
Let $$I_{\Z^*_+}: \Z^*_+ \to \Z^*_+$$ be the identity function:
 * $$\forall n \in \Z^*_+: I_{\Z^*_+} \left({n}\right) = n$$.

Then $$I_{\Z^*_+}$$ is completely multiplicative.

Proof
$$\forall m, n \in \Z^*_+: I_{\Z^*_+} \left({m n}\right) = m n = I_{\Z^*_+} \left({m}\right) I_{\Z^*_+} \left({n}\right)$$