Lemma 7.30.5. Let $\mathcal{C}$ be a site. Let $\mathcal{F} = h_ U^\#$ for some object $U$ of $\mathcal{C}$. Then $j_\mathcal {F} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ constructed in Lemma 7.30.1 agrees with the morphism of topoi $j_ U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ constructed in Section 7.25 via the identification $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\#$ of Lemma 7.25.4.

Proof. We have seen in Lemma 7.25.4 that the composition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is $j_{U!}$. The functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is $j_{\mathcal{F}!}$ by Lemma 7.30.1. Hence $j_{\mathcal{F}!} = j_{U!}$ via the identification. So $j_\mathcal {F}^{-1} = j_ U^{-1}$ (by adjointness) and so $j_{\mathcal{F}, *} = j_{U, *}$ (by adjointness again). $\square$

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