The Stacks project

Lemma 66.26.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A sheaf $\mathcal{F}$ of $\mathcal{O}_ X$-modules is given by the following data:

  1. for every $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ a sheaf $\mathcal{F}_ U$ of $\mathcal{O}_ U$-modules on $U_{\acute{e}tale}$,

  2. for every $f : U' \to U$ in $X_{\acute{e}tale}$ an isomorphism $c_ f : f_{small}^*\mathcal{F}_ U \to \mathcal{F}_{U'}$.

These data are subject to the condition that given any $f : U' \to U$ and $g : U'' \to U'$ in $X_{\acute{e}tale}$ the composition $c_ g \circ g_{small}^*c_ f$ is equal to $c_{f \circ g}$.

Proof. Combine Lemmas 66.26.1 and 66.18.13, and use the fact that any morphism between objects of $X_{\acute{e}tale}$ is an étale morphism of schemes. $\square$

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