Lemma 77.19.5. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-equivalence relation over $B$. For a scheme $S'$ over $S$ and $a, b \in U(S')$ the following are equivalent:

$a$ and $b$ map to the same element of $(U/R)(S')$, and

there exists an fppf covering $\{ f_ i : S_ i \to S'\} $ of $S'$ and morphisms $r_ i : S_ i \to R$ such that $a \circ f_ i = s \circ r_ i$ and $b \circ f_ i = t \circ r_ i$.

In other words, in this case the map of sheaves

is surjective.

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