Corollary

For every $a\in B$, the mapping $(\sigma_{\mathscr{F}})_a:\mathscr{F}_a\rightarrow \widetilde{\mathscr{F}}_a$ is bijective.

Since $j_{\mathscr{F}}$ is a B-isomorphism, the same is true of $E(\sigma_{\mathscr{F}})$, and $(\sigma_{\mathscr{F}})_a$ is deduced from it by passing to the fibres at $a$.

Let $\mathscr{G}$ be a presheaf on B relative to $\mathscr{B}$ and let $\varphi :\mathscr{F}\rightarrow \mathscr{G}$ be a morphism of presheaves. One denotes by $\widetilde{\varphi}:\widetilde{\mathscr{F}}\rightarrow \widetilde{\mathscr{G}}$ the morphism of sheaves $\mathscr{S}_{E(\varphi)}($I, p. 48, example 1), where $E(\varphi ): E_{\mathscr{F}}\rightarrow E_{\mathscr{G}}$ is the B-morphism associated with $\varphi$. For every open set $U\in \mathscr{B}$ and every $s\in \mathscr{F}(U)$, one has, by definition,

One has therefore:

(1) $\widetilde{\varphi}_U\circ \sigma_{\mathscr{F}}(U) =\sigma_{\mathscr{G}}(U)\circ \varphi_U$, for every $U\in \mathscr{B}$. In other words: