Profunctors

Quoting nCatLab: If C and D are categories, a profunctor from C to D is a functor Dᵒᵖ × C → Set. Such a profunctor is usually written as F : C ⇸ D. Every functor f : C → D induces two profunctors D(1, f) : C ⇸ D and D(f, 1) : D ⇸ C, defined by D(1,f)(d,c) = D(d, f(c)) and D(f, 1)(c,d) = D(f(c), d). These profunctors are called representable (or sometimes one of them is corepresentable). In particular the identity profunctor Id : C ⇸ C is represented by the identity functor and hence is given by the hom-functor C(−, −) : Cᵒᵖ × C → Set. The notion generalizes to many other kinds of categories. For instance, if C and D are enriched over some symmetric closed monoidal category V, then a profunctor from C to D is a V-functor Dᵒᵖ ⊗ C → V. If they are internal categories, then a profunctor C ⇸ D is an internal diagram on Dᵒᵖ × C, and so on. There are also other equivalent definitions in each case; see below. A profunctor is also sometimes called a (bi)module or a distributor or a correspondence, though the latter word is also used for a span. The term “module” tends to be common in Australia, especially in the enriched case; here the intuition is that for one-object V-categories, i.e. monoids in V, profunctors really are the same as bimodules between such monoids in the usual sense. “Profunctor” is perhaps more common in the Set-based and internal cases (but is also used in the enriched case); here the intuition is that a profunctor is a generalization of a functor, via the construction of “representable” profunctors. Jean Bénabou, who invented the term and originally used “profunctor,” now prefers “distributor,” which is supposed to carry the intuition that a distributor generalizes a functor in a similar way to how a distribution generalizes a function. Note that the convention that a profunctor is a functor Dᵒᵖ × C → Set is not universal; some authors reverse C and D and/or put the “op” on the other one. See the discussion below.

We capitalize Profunctor just like we capitalize Functor.