The Stacks project

Lemma 85.3.4. In Situation 85.3.3 there is an equivalence of categories between

  1. $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$, and

  2. the category of systems $(\mathcal{F}_ n, \mathcal{F}(\varphi ))$ described above.

In particular, the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ only depends on the topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n)$ and the morphisms of topoi $f_\varphi $.

Proof. See discussion above. $\square$

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