Lemma 87.34.3. Let S be a scheme, and let f : X \to Y be a morphism of formal algebraic spaces over S. Assume f is representable by algebraic spaces and étale. In this case there is a cocontinuous functor j : X_{\acute{e}tale}\to Y_{\acute{e}tale}. The morphism of topoi f_{small} is the morphism of topoi associated to j, see Sites, Lemma 7.21.1. Moreover, j is continuous as well, hence Sites, Lemma 7.21.5 applies.
Proof. This will follow immediately from the case of algebraic spaces (Properties of Spaces, Lemma 66.18.11) if we can show that the induced morphism X_{red} \to Y_{red} is étale. Observe that X \times _ Y Y_{red} is an algebraic space, étale over the reduced algebraic space Y_{red}, and hence reduced itself (by our definition of reduced algebraic spaces in Properties of Spaces, Section 66.7. Hence X_{red} = X \times _ Y Y_{red} as desired. \square
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