The Stacks project

Lemma 18.22.3. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$, let $\mathcal{F}$ be a sheaf on $\mathcal{C}$, and let $s : \mathcal{F} \to f^{-1}\mathcal{G}$ a morphism of sheaves. There exists a commutative diagram of ringed topoi

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \ar[rr]_{(j_\mathcal {F}, j_\mathcal {F}^\sharp )} \ar[d]_{(f_ c, f_ c^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal {G}) \ar[rr]^{(j_\mathcal {G}, j_\mathcal {G}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}'). } \]

The morphism $(f_ s, f_ s^\sharp )$ is equal to the composition of the morphism

\[ (f', (f')^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/f^{-1}\mathcal{G}, \mathcal{O}_{f^{-1}\mathcal{G}}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/{\mathcal{G}}, \mathcal{O}'_\mathcal {G}) \]

of Lemma 18.22.1 and the morphism

\[ (j, j^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/f^{-1}\mathcal{G}, \mathcal{O}_{f^{-1}\mathcal{G}}) \]

of Lemma 18.21.4. Given any morphisms $b : \mathcal{G}' \to \mathcal{G}$, $a : \mathcal{F}' \to \mathcal{F}$, and $s' : \mathcal{F}' \to f^{-1}\mathcal{G}'$ such that

\[ \xymatrix{ \mathcal{F}' \ar[r]_-{s'} \ar[d]_ a & f^{-1}\mathcal{G}' \ar[d]^{f^{-1}b} \\ \mathcal{F} \ar[r]^-s & f^{-1}\mathcal{G} } \]

commutes, then the following diagram of ringed topoi

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}', \mathcal{O}_{\mathcal{F}'}) \ar[rr]_{(j_{\mathcal{F}'/\mathcal{F}}, j_{\mathcal{F}'/\mathcal{F}}^\sharp )} \ar[d]_{(f_{s'}, f_{s'}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \ar[d]^{(f_ s, f_ s^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}', \mathcal{O}'_{\mathcal{G}'}) \ar[rr]^{(j_{\mathcal{G}'/\mathcal{G}}, j_{\mathcal{G}'/\mathcal{G}}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}, \mathcal{O}'_{\mathcal{G}'}) } \]


Proof. On the level of morphisms of topoi this is Sites, Lemma 7.31.3. To check that the diagrams commute as morphisms of ringed topoi use the commutative diagrams of Lemmas 18.21.4 and 18.22.1. $\square$

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