Remark 96.10.2. The constructions in Lemma 96.10.1 are compatible with étale localization. Here is a precise formulation. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by algebraic spaces $F$, $G$, and that the induced morphism $f : F \to G$ of algebraic spaces is étale. Denote $f_{small} : F_{\acute{e}tale}\to G_{\acute{e}tale}$ the corresponding morphism of ringed topoi. Then
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}), \mathcal{O}_ F) \ar[rr]_{f_{small}} \ar[d]_{i_ F} & & (\mathop{\mathit{Sh}}\nolimits (G_{\acute{e}tale}), \mathcal{O}_ G) \ar[d]^{i_ G} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale}), \mathcal{O}_\mathcal {X}) \ar[d]_{\pi _ F} \ar[rr]_ f & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{\acute{e}tale}), \mathcal{O}_\mathcal {Y}) \ar[d]^{\pi _ G} \\ (\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}), \mathcal{O}_ F) \ar[rr]^{f_{small}} & & (\mathop{\mathit{Sh}}\nolimits (G_{\acute{e}tale}), \mathcal{O}_ G) } \]
is a commutative diagram of ringed topoi. We omit the details.
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