Lurie's Unstraightening as a weak biequivalence of $\infty$-cosmoses

Abstract

We give a direct proof of the fact that Lurie's Unstraightening functor induces an equivalence between the strict $(\infty ,2)$-category of indexed quasi-categories and the strict $(\infty ,2)$-category of fibered quasi-categories over any given quasi-categorical base. We conclude that Unstraightening preserves simplicial cotensors up to a (strictly) natural homotopy equivalence, and thus gives rise to an accordingly weakened notion of cosmological biequivalence between the two underlying $\infty$-cosmoses.