The Stacks project

Lemma 76.8.2. With assumption and notation as in Theorem 76.8.1 the equivalence of categories $Y_{spaces, {\acute{e}tale}} \to X_{spaces, {\acute{e}tale}}$ restricts to equivalences of categories $Y_{\acute{e}tale}\to X_{\acute{e}tale}$ and $Y_{affine, {\acute{e}tale}} \to X_{affine, {\acute{e}tale}}$.

Proof. This is just the statement that given an object $V \in Y_{spaces, {\acute{e}tale}}$ we have $V$ is a(n affine) scheme if and only if $V \times _ Y X$ is a(n affine) scheme. Since $V \times _ Y X \to V$ is integral, universally injective, and surjective (as a base change of $X \to Y$) this follows from Limits of Spaces, Lemma 70.15.4 and Proposition 70.15.2. $\square$

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