Lemma 4.43. ([Uemura 2025, Lemma 4.2])
Let \(u\colon E \to B\) be a family in a topos \(T\). The functor
\[\Fam(T)_{/u} \longrightarrow T_{/B}\]
sending a pullback square \(v \to u\) to the induced morphism from the base of \(v\) to \(B\) is an equivalence.
Proof
The inverse sends a morphism \(f\colon X \to B\) to the pullback family \(f^*u\). Since morphisms in \(\Fam(T)\) are by definition pullback squares, these two constructions are inverse to each other.
References
- Taichi Uemura. Colimits in the ∞-category of ∞-topoi and étale morphisms. 2025.