Proposition 11

If the set I is finite, or if the space B is locally connected, the canonical morphism $\psi : F\rightarrow \prod F_i$ is an isomorphism.

Let U be an open set of B. It is clear that the mapping $\psi_U$ is injective. Let us show that it is surjective. It is a matter of proving that for every family $(f_i)_{i\in I}$ where $f_i$ is a locally constant mapping of U into $F_i$, and every point $a$ of U, there exists a neighbourhood V of $a$ in U such that for every $i\in I$, the mapping $f_i|V$ is constant. When the set I is finite, the existence of such a neighbourhood is clear. When the space B is locally connected, it is enough to take for V a connected neighbourhood of $a$ in U. This proves the proposition.