On the straightening of every functor

We show that any functor between ∞-categories can be straightened. More precisely, we show that for any ∞-category C, there is an equivalence between the ∞-category (Cat∞)/C of ∞-categories over C and the ∞-category of unital lax functors from C to the double ∞-category Corr of correspondences. The proof relies on a certain universal property of the Morita category which is of independent interest.