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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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