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_) /₂ 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.