The Stacks project

Remark 7.29.8. Notation and assumptions as in Lemma 7.29.6. Suppose that in addition the original morphism of topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ is an equivalence. Then the construction in the proof of Lemma 7.29.6 gives two functors

\[ \mathcal{C} \rightarrow \mathcal{C}' \leftarrow \mathcal{D} \]

which are both special cocontinuous functors. Hence in this case we can actually factor the morphism of topoi as a composition

\[ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \rightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') = \mathop{\mathit{Sh}}\nolimits (\mathcal{D}') \leftarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \]

as in Remark 7.29.7, but with the middle morphism an identity.

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