The best one can hope is therefore to build a bicategory Prof whose  A functor formula_53 can be seen as a profunctor formula_54 by postcomposing with the Yoneda functor:  It can be shown that such a profunctor formula_56 has a right adjoint. Moreover, this is a characterization: a profunctor formula_4 has a right adjoint if and only if formula_58 factors through the Cauchy completion of formula_3, i.e. there exists a functor formula_53 such that formula_61.