It is well known that a distributor/profunctor $A \not\rightarrow B$, i.e. a functor $B^{\text{op}} \times A \to \mathrm{Set}$, is equivalent to a two-sided discrete fibration from $A$ to $B$. Furthermore, this correspondence extends to an equivalence of categories $\mathrm{Dist}(A, B) \simeq \mathrm{DFib}(A, B)$ pseudonatural in $A$ and $B$ (this is mentioned without proof on page 177 of Street's Elementary cosmoi I, for instance; and with a sketch of a proof as Theorem 2.3.2 of Loregian and Riehl's Categorical notions of fibration).
Evidently this correspondence should extend to a biequivalence of bicategories $\mathrm{Dist} \sim \mathrm{DFib}$. However, I have been unable to find an explicit proof of this fact in the literature. Does a reference exist?
