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

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

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

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