Lemma 66.18.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

The continuous functor

\[ Y_{spaces, {\acute{e}tale}} \longrightarrow X_{spaces, {\acute{e}tale}}, \quad V \longmapsto X \times _ Y V \]induces a morphism of sites

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

The morphism of topoi associated to $f_{spaces, {\acute{e}tale}}$ induces, via Lemma 66.18.3, 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.

If $f$ is a representable morphism of algebraic spaces, then $f_{small}$ comes from a morphism of sites $X_{\acute{e}tale}\to Y_{\acute{e}tale}$, corresponding to the continuous functor $V \mapsto X \times _ Y V$.

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