Lemma 96.12.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \textit{Mod}(\mathcal{Y}_{\acute{e}tale}, \mathcal{O}_\mathcal {Y}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ preserves locally quasi-coherent sheaves.
Proof. Let $\mathcal{G}$ be locally quasi-coherent on $\mathcal{Y}$. Choose an object $x$ of $\mathcal{X}$ lying over the scheme $U$. The restriction $x^*f^*\mathcal{G}|_{U_{\acute{e}tale}}$ equals $(f \circ x)^*\mathcal{G}|_{U_{\acute{e}tale}}$ hence is a quasi-coherent sheaf by assumption on $\mathcal{G}$. $\square$
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