The Stacks project

Lemma 39.20.3. In the situation of Definition 39.20.1. Assume there is a scheme $M$, and a morphism $U \to M$ such that

  1. the morphism $U \to M$ equalizes $s, t$,

  2. the morphism $U \to M$ induces a surjection of sheaves $h_ U \to h_ M$ in the $\tau $-topology, and

  3. the induced map $(t, s) : R \to U \times _ M U$ induces a surjection of sheaves $h_ R \to h_{U \times _ M U}$ in the $\tau $-topology.

In this case $M$ represents the quotient sheaf $U/R$.

Proof. Condition (1) says that $h_ U \to h_ M$ factors through $U/R$. Condition (2) says that $U/R \to h_ M$ is surjective as a map of sheaves. Condition (3) says that $U/R \to h_ M$ is injective as a map of sheaves. Hence the lemma follows. $\square$

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