Proposition 5

Let X and Y be topological spaces and let $f: X\rightarrow$ Y be a continuous and relatively connected mapping. For every pair $(T,T')$ of coverings of Y, the mapping $f^*:\mathscr{C}_Y(T; T')\rightarrow$ $\mathscr{C}_X(X\times_YT; X\times_YT')$ is bijective.

Let $\mathscr{F}$ be the sheaf on X of X-morphisms from $X\times_YT$ into $X\times_YT'$ and let $\mathscr{G}$ be the sheaf on Y of Y-morphisms from T into $T'($I, p. 45, example 4). For every open set U of Y, let us put $\varphi_U= (f_U)^*:\mathscr{G}(U)\rightarrow$

$\mathscr{F}(\overset{-1}{f}(U))$. The mappings $\varphi_U$ define a sheaf morphism $\varphi :\mathscr{G}\rightarrow \varphi_*(\mathscr{F})$ and it is enough to prove that $\varphi$ is an isomorphism of sheaves.

Since Y is locally connected (IV, p. 354, cor. 2), the connected open sets over which T and $T'$ are trivializable form a basis of the topology of Y. By corollary 2 of I, p. 55, it is enough to prove that for such an open set U, the mapping $\varphi_U$ is bijective, which allows us to suppose that Y is connected and that the coverings T and $T'$ are the trivial coverings $Y\times F$ and $Y\times F'$ where F and $F'$ are sets endowed with the discrete topology. The mapping $(x,(y, t))\mapsto (x, t)$ identifies the X-space $X\times_Y(Y\times F)$ with $X\times F$ (resp. the X-space $X\times_Y(Y\times F')$ with $X\times F'$). Since the space X is connected (IV, p. 354, cor. 2), the sets $\mathscr{C}_Y(Y\times F; Y\times F')$ and $\mathscr{C}_X(X\times F; X\times F')$ are both identified with the set $\mathscr{F}(F; F')$ of mappings from F into $F'$, and the mapping $f^*$ is identified with the identity mapping of $\mathscr{F}(F; F')$. This concludes the proof.