Profunctor

Summary

A profunctor, also called a distributor or bimodule, from a category $\mathcal{C}$ to a category $\mathcal{D}$ is a functor $\mathcal{D}^{\mathrm{op}} \times \mathcal{C} \to \mathbf{Set}$. It generalizes functors: every functor gives a profunctor. It also generalizes relations: a relation between sets is a profunctor between discrete categories with values in truth values. Profunctors compose by a coend, and categories, profunctors and natural transformations form a bicategory. They are the categorical analogue of bimodules between rings.