Lemma 7.30.1. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ is a topos. There is a canonical morphism of topoi
\[ j_\mathcal {F} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]
which is a localization as in Section 7.25 such that
the functor $j_\mathcal {F}^{-1}$ is the functor $\mathcal{H} \mapsto \mathcal{H} \times \mathcal{F}/\mathcal{F}$, and
the functor $j_{\mathcal{F}!}$ is the forgetful functor $\mathcal{G}/\mathcal{F} \mapsto \mathcal{G}$.
Proof.
Apply Lemma 7.29.5. This means we may assume $\mathcal{C}$ is a site with subcanonical topology, and $\mathcal{F} = h_ U = h_ U^\# $ for some $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Hence the material of Section 7.25 applies. In particular, there is an equivalence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ such 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 equal to $j_{U!}$, see Lemma 7.25.4. Denote $a : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$ the inverse functor, so $j_{\mathcal{F}!} = j_{U!} \circ a$, $j_\mathcal {F}^{-1} = a^{-1} \circ j_ U^{-1}$, and $j_{\mathcal{F}, *} = j_{U, *} \circ a$. The description of $j_{\mathcal{F}!}$ follows from the above. The description of $j_\mathcal {F}^{-1}$ follows from Lemma 7.25.7.
$\square$
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