Old - Monoidal profunctor

Definition. A monoidal profunctor between two monoidal categories $C$ and $\mathbb{D}$ is a profunctor $P:C\to \mathbb{D}$ between the underlying categories, endowed with maps $(⊠):P(X;Y)\times P(X^{\prime },Y^{\prime })\to P(X\otimes X^{\prime };Y\otimes Y^{\prime }),$ $n:1\to P(I;I),$ that are natural and that moreover satisfy

• associativity, $\alpha >(p⊠(q⊠r))<\alpha =((p⊠q)⊠r)$,
• left unitality, $\lambda >(n⊠p)<\lambda =p$,
• right unitality, $\rho >(p⊠n)<\rho =p$.

Characterizations