Lemma 64.9.1. Let $F$ be an algebraic space over $S$. Let $f : U \to F$ be a surjective étale morphism from a scheme to $F$. Set $R = U \times _ F U$. Then

$j : R \to U \times _ S U$ defines an equivalence relation on $U$ over $S$ (see Groupoids, Definition 39.3.1).

the morphisms $s, t : R \to U$ are étale, and

the diagram

\[ \xymatrix{ R \ar@<1ex>[r] \ar@<-1ex>[r] & U \ar[r] & F } \]is a coequalizer diagram in $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})$.

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