KöMaL 2017 A706

Let $\mathbb{Z}^+$ denote the set of positive integers. Find all functions $f\colon \mathbb{Z}^+\to\mathbb{Z}^+$ which satisfy the following: $\bullet$ $f(mn)=f(m)f(n)$ for all $m,n\in\mathbb{Z}^+$, and $\bullet$ $f^{(n)}(n)=n$ for all $n\in\mathbb{Z}^+$ (in other words, $f\Big(f\big(\dots \big(f(n)\big)\dots\big)\!\Big)=n$, where there are $n$ pairs of brackets on the left-hand side).