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

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

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

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