Corollary 2

Let $(\mathscr{F}_i)_{i\in I}$ be a family of sheaves on B. Suppose that the space B is locally connected and that every point of B possesses an open neighbourhood U such that, for every $i\in I$, the sheaf $\mathscr{F}_i|U$ is isomorphic to a constant sheaf. Then the product sheaf $\prod\mathscr{F}_i$ is locally constant.