Lemma 87.12.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$ which is representable by algebraic spaces and smooth (for example étale). Then $X_{red} = X \times _ Y Y_{red}$.
Proof. (The étale case follows directly from the construction of the underlying reduced algebraic space in the proof of Lemma 87.12.1.) Assume $f$ is smooth. Observe that $X \times _ Y Y_{red} \to Y_{red}$ is a smooth morphism of algebraic spaces. Hence $X \times _ Y Y_{red}$ is a reduced algebraic space by Descent on Spaces, Lemma 74.9.5. Then the universal property of reduction shows that the canonical morphism $X_{red} \to X \times _ Y Y_{red}$ is an isomorphism. $\square$
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