The Stacks project

Lemma 18.21.3. With $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ and $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ as in Lemma 18.21.1. If $\mathcal{F} = h_ U^\# $ for some object $U$ of $\mathcal{C}$ then via the identification $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ of Sites, Lemma 7.25.4 we have

  1. canonically $\mathcal{O}_ U = \mathcal{O}_\mathcal {F}$, and

  2. with these identifications we have $(j_\mathcal {F}, j_\mathcal {F}^\sharp ) = (j_ U, j_ U^\sharp )$.

Proof. The assertion for underlying topoi is Sites, Lemma 7.30.5. Note that $\mathcal{O}_ U$ is the restriction of $\mathcal{O}$ which by Sites, Lemma 7.25.7 corresponds to $\mathcal{O} \times h_ U^\# $ under the equivalence of Sites, Lemma 7.25.4. By definition of $\mathcal{O}_\mathcal {F}$ we get (1). What's left is to prove that $j_\mathcal {F}^\sharp = j_ U^\sharp $ under this identification. We omit the verification. $\square$

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