Lemma 39.20.4. Let $\tau \in \{ Zariski, {\acute{e}tale}, fppf, smooth, syntomic\} $. Let $S$ be a scheme. Let $j : R \to U \times _ S U$ be a pre-equivalence relation over $S$. Assume $U, R, S$ are objects of a $\tau $-site $\mathit{Sch}_\tau $. For $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_\tau )$ and $a, b \in U(T)$ the following are equivalent:

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

there exists a $\tau $-covering $\{ f_ i : T_ i \to T\} $ of $T$ and morphisms $r_ i : T_ i \to R$ 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 $\tau $-sheaves

is surjective.

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