Promonoidal categories are the biased version of coherent multicategories. Promonoidal categories are pseudomonoids in the monoidal bicategory of profunctors. Examples of promonoidal category include the promonoidal category of optics, the promonoidal category of spliced arrows the promonoidal category of shufflings or the multicategory of convex mixtures.
See also: profunctor, monoidal category, malleable multicategories are promonoidal, Proeffectful categories, Prostrong promonad, Prepromonoidality, Promonoidal contour, promonoidal functor, Forgetting promonoidality, Old - Promonoidal category.
References:
- On Closed Categories of Functors (Day), where promonoidal categories are called premonoidal categories.