The Stacks project

Lemma 66.28.5. Let $X$, $Y$ be algebraic spaces over $\mathbf{Z}$. If

\[ (g, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y) \]

is an isomorphism of ringed topoi, then there exists a unique morphism $f : X \to Y$ of algebraic spaces such that $(g, g^\sharp )$ is isomorphic to $(f_{small}, f^\sharp )$ and moreover $f$ is an isomorphism of algebraic spaces.

Proof. By Theorem 66.28.4 it suffices to show that $(g, g^\sharp )$ is a morphism of locally ringed topoi. By Modules on Sites, Lemma 18.40.8 (and since the site $X_{\acute{e}tale}$ has enough points) it suffices to check that the map $\mathcal{O}_{Y, q} \to \mathcal{O}_{X, p}$ induced by $g^\sharp $ is a local ring map where $q = f \circ p$ and $p$ is any point of $X_{\acute{e}tale}$. As it is an isomorphism this is clear. $\square$

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