Remark

Let $\mathscr{F}$ be a sheaf on B relative to the base $\mathscr{B}$. Let $B'$ be an open subset of B, let $\mathscr{B}'$ be a basis of the topology of $B'$ such that $\mathscr{B}'\subset \mathscr{B}$. Let $\mathscr{F}|\mathscr{B}'$ be the presheaf on $B'$ relative to the base $\mathscr{B}'$ deduced from $\mathscr{F}$ by restriction.

1) The set $X_{\mathscr{F}|\mathscr{B}'}$ is then a subset of $X_{\mathscr{F}}$ and the equivalence relation $R_{\mathscr{F}}$ induces in $X_{\mathscr{F}|\mathscr{B}'}$ the equivalence relation $R_{\mathscr{F}|\mathscr{B}'}$. Hence one deduces a canonical injection $i$ of $E_{\mathscr{F}|\mathscr{B}'}$ into $E_{\mathscr{F}}$. Its image is $\overset{-1}{p}(B')$ since for every element $[U, s, x]$ of $\overset{-1}{p}(B')$, there exists an element V of $\mathscr{B}'$ such that $x\in V$ and $V\subset U$, and one has $[U, s, x] =i([V, r_{VU}(s), x])$. The mapping $i$ is continuous because the topology of $X_{\mathscr{F}|\mathscr{B}'}$ is the finest making continuous the mappings defined by $x\mapsto [U, s, x]$, for $U\in \mathscr{B}'$ and $s\in \mathscr{F}(U)$. By corollary 2 of Proposition 6 in I, p. 30, the canonical injection $i$ of $E_{\mathscr{F}|\mathscr{B}'}$ into $E_{\mathscr{F}}$ induces a $B'$-isomorphism of $E_{\mathscr{F}|\mathscr{B}'}$ onto $\overset{-1}{p}(B')$.

In particular, when $B'$ is equal to B$,i: E_{\mathscr{F}|\mathscr{B}'}\rightarrow E_{\mathscr{F}}$ is a B-isomorphism of étalé spaces.

2) Let $\mathscr{G}$ be a presheaf on B relative to the base $\mathscr{B}$ and let $\varphi :\mathscr{F}\rightarrow \mathscr{G}$ be a morphism of presheaves. The family $\varphi '= (\varphi_U)_{U\in\mathscr{B}'}$ is a morphism of presheaves from $\mathscr{F}|\mathscr{B}'$ into $\mathscr{G}|\mathscr{B}'$. The diagram

$ii'$

$E_{\mathscr{F}}^{E(\varphi)}E_{\mathscr{G}}$ , where $i$ and $i'$ are the canonical injections, is commutative.