The Stacks project

Lemma 87.34.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$.

  1. There is a continuous functor $Y_{spaces, {\acute{e}tale}} \to X_{spaces, {\acute{e}tale}}$ which induces a morphism of sites

    \[ f_{spaces, {\acute{e}tale}} : X_{spaces, {\acute{e}tale}} \to Y_{spaces, {\acute{e}tale}}. \]

  2. The rule $f \mapsto f_{spaces, {\acute{e}tale}}$ is compatible with compositions, in other words $(f \circ g)_{spaces, {\acute{e}tale}} = f_{spaces, {\acute{e}tale}} \circ g_{spaces, {\acute{e}tale}}$ (see Sites, Definition 7.14.5).

  3. The morphism of topoi associated to $f_{spaces, {\acute{e}tale}}$ induces, via (87.34.1.1), a morphism of topoi $f_{small} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ whose construction is compatible with compositions.

Proof. The only point here is that $f$ induces a morphism of reductions $X_{red} \to Y_{red}$ by Lemma 87.12.1. Hence this lemma is immediate from the corresponding lemma for morphisms of algebraic spaces (Properties of Spaces, Lemma 66.18.8). $\square$

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