  Let $FI$ be a skeleton of the category of finite sets and injective maps, and
$FI^m$ the product of $m$ copies of $FI$. We prove that if an $FI^m$-module is
generated in degree $\leqslant d$ and related in degree $\leqslant r$, then its
regularity is bounded above by a function of $m$, $d$, and $r$.
