Remark 1

Let $(E_i, p_i)_{i\in I}$ be a family of coverings of the topological space B. Suppose that the space B is locally connected and that there exists an open covering $(U_j)_{j\in J}$ of B such that, for every $j\in J$ and every $i\in I$, the covering $E_i$ is trivializable over $U_j$. For $i\in I$, let $\mathscr{F}_i$ be the locally constant sheaf of sections of $E_i$ and let $\mathscr{F}$ be the product sheaf $\prod_i\mathscr{F}_i$. By the preceding corollary, $\mathscr{F}$ is a locally constant sheaf and the étalé B-space E associated with it is therefore a covering (I, p. 86, prop. 8). For $i\in I$, let pr$_i: E\rightarrow E_i$ be the B-morphism induced by the projection morphism of index $i,\mathscr{F}\rightarrow \mathscr{F}_i$.

Now consider a covering $(Y, q)$ of B and, for every $i\in$ I, let $f_i: Y\rightarrow E_i$ be a B-morphism. If $\mathscr{G}$ denotes the locally constant sheaf of sections of $q$, the B-morphisms $f_i$ induce sheaf morphisms $\widetilde{f}_i:\mathscr{G}\rightarrow \mathscr{F}_i$. Let $\widetilde{f}:\mathscr{G}\rightarrow \mathscr{F}$ be the unique sheaf morphism such that $\widetilde{f}_i=$ pr$_i\circ \widetilde{f}$ for every $i\in I$. There then exists a unique B-morphism $f: Y\rightarrow E$ such that $f_i$ = pr$_i\circ f$ for every $i:$ this is the B-morphism which induces $\widetilde{f}$.

One sometimes says that E is the product covering of the family $(E_i)_{i\in I}$.