Lemma 96.25.2. Let p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf} be a category fibred in groupoids. Let \mathcal{F} be an \mathcal{O}-module on \mathcal{X}_{affine}. The following are equivalent
for every morphism x \to x' of \mathcal{X}_{affine} such that p(x) \to p(x') is an étale morphism (of affine schemes), the map \mathcal{F}(x') \otimes _{\mathcal{O}(x')} \mathcal{O}(x) \to \mathcal{F}(x) is an isomorphism,
\mathcal{F} is a sheaf for the étale topology on \mathcal{X}_{affine} and for every object x of \mathcal{X}_{affine} the restriction x^*\mathcal{F}|_{U_{affine, {\acute{e}tale}}} is quasi-coherent where U = p(x),
\mathcal{F} corresponds to a locally quasi-coherent module on \mathcal{X} via the equivalence (96.24.3.1) for the étale topology.
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