Lemma 6.155.
Every topos is a pullback in \(\Topos\) of a diagram of presheaf topoi \(\PSh(D)\) with \(D\) admitting finite limits.
Proof sketch
Present the topos as a left exact localization of a presheaf topos. The localization is the equalizer of the identity and its idempotent localization functor. Replacing this equalizer by the usual pullback involving an arrow category expresses the original topos as a pullback of three presheaf topoi. Closing the small indexing categories under finite limits gives the stated form. See [Anel and Lejay 2018, Section 4, in particular the construction preceding Theorem 4.33].
References
- Mathieu Anel, Damien Lejay. Exponentiable higher toposes. 2018.