Proposition 10

If the set I is finite, the B-morphism Φ is an isomorphism.

By the corollary of I, p. 32, the B-space $E'$ is étalé, and it is enough to prove that the B-morphism Φ is bijective ( I, p. 30, Cor. 2). For every point $b$ of B, the restriction $\Phi_b: E_b\rightarrow E'_b$ of Φ to the fibres at $b$ is identified with the canonical mapping from lim$\longrightarrow \prod^{i\in I}\mathscr{F}_i(U)$ into $\prod_{i\in I}$ lim$\longrightarrow \mathscr{F}_i(U)$, where U runs through the set of open sets of B which contain $b($cf. I, p. 50). By Prop. 10 of E, III, p. 67, this mapping is a bijection.

Now consider the case of locally constant sheaves on a topological space B. Let $(F_i)_{i\in I}$ be a family of sets and put $F =\prod_{i\in I}F_i$. A canonical morphism $\psi = (\psi_U)$ of the constant sheaf F into the product $\prod_{i\in I}F_i$ of the constant sheaves $F_i$ is defined by putting, for every open set U of B and every locally constant function $f: U\rightarrow F,\psi_U(f) =$ (pr$_i\circ f$)$_{i\in I}$.