  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.
