Lemma 94.11.4. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of modules on $\mathcal{X}$. The following are equivalent

$\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$ and $\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$ in the sense of Modules on Sites, Definition 18.23.1,

$\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ and $\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ in the sense of Modules on Sites, Definition 18.23.1, and

$\mathcal{F}$ is a quasi-coherent module on $\mathcal{X}$ in the sense of Definition 94.11.1.

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