Lemma 39.3.2. Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j : R \to U \times _ S U$ be a pre-relation. Let $g : U' \to U$ be a morphism of schemes. Finally, set

$R' = (U' \times _ S U')\times _{U \times _ S U} R \xrightarrow {j'} U' \times _ S U'$

Then $j'$ is a pre-relation on $U'$ over $S$. If $j$ is a relation, then $j'$ is a relation. If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation. If $j$ is an equivalence relation, then $j'$ is an equivalence relation.

Proof. Omitted. $\square$

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