The Stacks project

Lemma 66.29.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ is given by the following data:

1. for every $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ a quasi-coherent $\mathcal{O}_ U$-module $\mathcal{F}_ U$ 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 $c_{f \circ g}$.