definition

For \(P:\cal M \nrightarrow M\) monoidal profunctor and \(Q:\cal C \nrightarrow D\), define \(Q/P:\cal C \nrightarrow D\) as \[Q/P(C,D) = \int _{MM'} \int _{C'D'} \mathcal C(C', MC) \times P(M,M') \times \mathcal D(M'D, D') \to Q(C',D').\] Then it’s easy to see that \(-/\cal M \cong \Theta \), by using a couple of Yoneda reductions: \[Q/{\cal M}(C,D) = \int _{MM'} \int _{C'D'} \mathcal C(C', MC) \times {\cal M}(M,M') \times \mathcal D(M'D, D') \to Q(C',D')\\ \cong \int _{M} \int _{C'D'} \mathcal C(C', MC) \times \mathcal D(MD, D') \to Q(C',D')\\ \cong \int _{M} Q(MC,MD) = \Theta Q(C,D).\]