seems to be a useful general rule here:

\[ \Phi \text{ is a profunctor} \iff \mathcal{X}(x', x) \otimes \Phi(x, y) \otimes \mathcal{Y}(y, y') \leq \Phi(x', y') \text{ for all } x, x', y, y' \]

\[ \Phi \text{ is a profunctor} \iff \mathcal{X}(x', x) \otimes \Phi(x, y) \otimes \mathcal{Y}(y, y') \leq \Phi(x', y') \text{ for all } x, x', y, y' \]