  In this paper we study representation theory of the category FI$^m$
introduced by Gadish which is a product of copies of the category FI, and show
that quite a few interesting representational and homological properties of FI
can be generalized to FI$^m$ in a natural way. In particular, we prove the
representation stability property of finitely generated FI$^m$-modules over
fields of characteristic 0.
