The Stacks project

Lemma 65.10.1. Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \to U \times _ S U$ be an étale equivalence relation on $U$ over $S$. Let $U' \to U$ be an étale morphism. Let $R'$ be the restriction of $R$ to $U'$, see Groupoids, Definition 39.3.3. Then $j' : R' \to U' \times _ S U'$ is an étale equivalence relation also.

Proof. It is clear from the description of $s', t'$ in Groupoids, Lemma 39.18.1 that $s' , t' : R' \to U'$ are étale as compositions of base changes of étale morphisms (see Morphisms, Lemma 29.36.4 and 29.36.3). $\square$

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