Lemma 18.21.1. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a sheaf. For a sheaf $\mathcal{H}$ on $\mathcal{C}$ denote $\mathcal{H}_\mathcal {F}$ the sheaf $\mathcal{H} \times \mathcal{F}$ seen as an object of the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$. The pair $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F})$ is a ringed topos and there is a canonical morphism of ringed topoi

which is a localization as in Section 18.19 such that

the functor $j_\mathcal {F}^{-1}$ is the functor $\mathcal{H} \mapsto \mathcal{H}_\mathcal {F}$,

the functor $j_\mathcal {F}^*$ is the functor $\mathcal{H} \mapsto \mathcal{H}_\mathcal {F}$,

the functor $j_{\mathcal{F}!}$ on sheaves of sets is the forgetful functor $\mathcal{G}/\mathcal{F} \mapsto \mathcal{G}$,

the functor $j_{\mathcal{F}!}$ on sheaves of modules associates to the $\mathcal{O}_\mathcal {F}$-module $\varphi : \mathcal{G} \to \mathcal{F}$ the $\mathcal{O}$-module which is the sheafification of the presheaf

\[ V \longmapsto \bigoplus \nolimits _{s \in \mathcal{F}(V)} \{ \sigma \in \mathcal{G}(V) \mid \varphi (\sigma ) = s \} \]for $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

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