Lemma 83.5.2. Let $B \to S$ as in Section 83.2. Let $j : R \to U \times _ B U$ be a pre-equivalence relation of algebraic spaces over $B$. Then
\[ O_ u = \{ u' \in |U| \text{ such that } \exists r \in |R|, \ s(r) = u, \ t(r) = u'\} . \]
Proof. By the aforementioned Groupoids in Spaces, Lemma 78.4.4 we see that the orbits $O_ u$ as defined in the lemma give a disjoint union decomposition of $|U|$. Thus we see they are equal to the orbits as defined in Definition 83.5.1. $\square$
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