Lemma 96.23.1. Let S be a scheme. Let \mathcal{X} be an algebraic stack over S. Let \mathcal{F} be a presheaf of \mathcal{O}_\mathcal {X}-modules. Assume
\mathcal{F} is locally quasi-coherent, and
for any morphism \varphi : x \to y of \mathcal{X} which lies over a morphism of schemes f : U \to V which is flat and locally of finite presentation the comparison map c_\varphi : f_{small}^*\mathcal{F}|_{V_{\acute{e}tale}} \to \mathcal{F}|_{U_{\acute{e}tale}} of (96.9.4.1) is an isomorphism.
Then \mathcal{F} is a sheaf for the fppf topology.
Proof.
Let \{ x_ i \to x\} be an fppf covering of \mathcal{X} lying over the fppf covering \{ f_ i : U_ i \to U\} of schemes over S. By assumption the restriction \mathcal{G} = \mathcal{F}|_{U_{\acute{e}tale}} is quasi-coherent and the comparison maps f_{i, small}^*\mathcal{G} \to \mathcal{F}|_{U_{i, {\acute{e}tale}}} are isomorphisms. Hence the sheaf condition for \mathcal{F} and the covering \{ x_ i \to x\} is equivalent to the sheaf condition for \mathcal{G}^ a on (\mathit{Sch}/U)_{fppf} and the covering \{ U_ i \to U\} which holds by Descent, Lemma 35.8.1.
\square
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