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