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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