Corollary 1

A finite product of locally constant sheaves is locally constant.

Let $(\mathscr{F}_i)_{i\in I}$ be a finite family of locally constant sheaves on B. Every point $a$ of B possesses an open neighbourhood U in B such that, for every $i\in I$, the sheaf $\mathscr{F}_i|U$ is isomorphic to a constant sheaf. The same is then true of the sheaf $((\prod\mathscr{F}_i)|U$, which is equal to $\prod ((\mathscr{F}_i|U)$.