The Stacks project

Situation 21.38.3. Let $(\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a ringed site. Let $u : \mathcal{C}' \to \mathcal{C}$ be a $1$-morphism of fibred categories over $\mathcal{D}$ (Categories, Definition 4.33.9). Endow $\mathcal{C}$ and $\mathcal{C}'$ with their inherited topologies (Stacks, Definition 8.10.2) and let $\pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$, $\pi ' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$, and $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be the corresponding morphisms of topoi (Stacks, Lemma 8.10.3). Set $\mathcal{O}_\mathcal {C} = \pi ^{-1}\mathcal{O}_\mathcal {D}$ and $\mathcal{O}_{\mathcal{C}'} = (\pi ')^{-1}\mathcal{O}_\mathcal {D}$. Observe that $g^{-1}\mathcal{O}_\mathcal {C} = \mathcal{O}_{\mathcal{C}'}$ so that

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[rd]_{\pi '} \ar[rr]_ g & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \ar[ld]^\pi \\ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) } \]

is a commutative diagram of morphisms of ringed topoi.

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