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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