The Stacks project

Lemma 96.9.3. Let $\mathcal{F}$ be an étale sheaf on $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$.

1. If $\varphi : x \to y$ and $\psi : y \to z$ are morphisms of $\mathcal{X}$ lying over $a : U \to V$ and $b : V \to W$, then the composition

   \[ a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_{\acute{e}tale}})) \xrightarrow {a_{small}^{-1}c_\psi } a_{small}^{-1}(\mathcal{F}|_{V_{\acute{e}tale}}) \xrightarrow {c_\varphi } \mathcal{F}|_{U_{\acute{e}tale}} \]

   is equal to $c_{\psi \circ \varphi }$ via the identification

   \[ (b \circ a)_{small}^{-1}(\mathcal{F}|_{W_{\acute{e}tale}}) = a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_{\acute{e}tale}})). \]

2. If $\varphi : x \to y$ lies over an étale morphism of schemes $a : U \to V$, then (96.9.2.2) is an isomorphism.

3. Suppose $f : \mathcal{Y} \to \mathcal{X}$ is a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ and $y$ is an object of $\mathcal{Y}$ lying over the scheme $U$ with image $x = f(y)$. Then there is a canonical identification $f^{-1}\mathcal{F}|_{U_{\acute{e}tale}} = \mathcal{F}|_{U_{\acute{e}tale}}$.

4. Moreover, given $\psi : y' \to y$ in $\mathcal{Y}$ lying over $a : U' \to U$ the comparison map $c_\psi : a_{small}^{-1}(f^{-1}\mathcal{F}|_{U_{\acute{e}tale}}) \to f^{-1}\mathcal{F}|_{U'_{\acute{e}tale}}$ is equal to the comparison map $c_{f(\psi )} : a_{small}^{-1}\mathcal{F}|_{U_{\acute{e}tale}} \to \mathcal{F}|_{U'_{\acute{e}tale}}$ via the identifications in (3).