On the straightening of every functor

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