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The Stacks project

Lemma 18.21.4. Let (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) be a ringed topos. If s : \mathcal{G} \to \mathcal{F} is a morphism of sheaves on \mathcal{C} then there exists a natural commutative diagram of morphisms of ringed topoi

\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{G}, \mathcal{O}_\mathcal {G}) \ar[rd]_{(j_\mathcal {G}, j_\mathcal {G}^\sharp )} \ar[rr]_{(j, j^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \ar[ld]^{(j_\mathcal {F}, j_\mathcal {F}^\sharp )} \\ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) & }

where (j, j^\sharp ) is the localization morphism of the ringed topos (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) at the object \mathcal{G}/\mathcal{F}.

Proof. All assertions follow from Sites, Lemma 7.30.6 except the assertion that j_\mathcal {G}^\sharp = j^\sharp \circ j^{-1}(j_\mathcal {F}^\sharp ). We omit the verification. \square


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