Let $d(n)$ be the number of positive divisors of a positive integer $n$.  Let $\mathbb{N}$ be the set of all positive integers.  Say that a function $F$ from $\mathbb{N}$ to $\mathbb{N}$ is [i]divisor-respecting[/i] if $d(F(mn)) = d(F(m)) d(F(n))$ for all positive integers $m$ and $n$, and $d(F(n)) \le d(n)$ for all positive integers $n$.  Find all divisor-respecting functions.  Justify your answer.