Remark 2

With the preceding notations, the canonical B-morphism $\Phi : E\rightarrow \prod_BE_i$ is bijective. The question is in fact of a local nature on B, and one may suppose that, for every $i\in I$, the B-space $E_i$ is isomorphic to the B-space $(B\times F_i$, pr$_1)$, where $F_i$ is a discrete topological space, so that the sheaf $\mathscr{F}_i$ is isomorphic to the sheaf $F_i$. By Prop. 11, the sheaf $\mathscr{F}$ is identified with the sheaf F, where F is the set $\prod F_i$, endowed with the discrete topology, and the mapping Φ is identified with the canonical mapping $B\times F\rightarrow B\times (\prod_iF_i)$, which is bijective.