Lemma 39.18.3. The notions of restricting groupoids and (pre-)equivalence relations defined in Definitions 39.18.2 and 39.3.3 agree via the constructions of Lemmas 39.13.2 and 39.13.3.
Proof. What we are saying here is that $R'$ of Lemma 39.18.1 is also equal to
\[ R' = (U' \times _ S U')\times _{U \times _ S U} R \longrightarrow U' \times _ S U' \]
In fact this might have been a clearer way to state that lemma. $\square$
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