A \(\mathcal{V}\) profunctor, denoted \(\mathcal{X}\overset{\phi}{\nrightarrow} \mathcal{Y}\) - where \(\mathcal{V}=(V,\leq,I,\otimes)\) is a (unital commutative) quantale, and \(\mathcal{X},\mathcal{Y}\) are \(\mathcal{V}\) categories.
A \(\mathcal{V}\) functor \(\mathcal{X}^{op}\times Y \xrightarrow{\phi} \mathcal{V}\)