nLab
Prof

Prof is the 2-category of categories, profunctors, and natural transformations. If profunctors are categorified binary relations, then Prof is a categorification of Rel.

Recall that a profunctor from A to B is a functor B op×ASet. Composition of profunctors in Prof is by the “tensor product of functors” coend construction: if H:AB and K:BC, their composite is given as a functor C op×ASet by

(c,a) bBH(b,a)×K(c,b).(c,a)\mapsto \int^{b\in B} H(b,a)\times K(c,b).

The identity on a category A is its hom-functor Hom A(,).

Note that as defined here, Prof is a weak 2-category or bicategory. A naturally defined equivalent strict 2-category has the same objects, but the morphisms AB are cocontinuous functors PAPB, where PA is the presheaf category of A. This is equivalent because a profunctor AB can equivalently be regarded as a functor APB, and PA is the free cocompletion of A.

Note that every functor f:AB gives two representable profunctors B(f,) and B(,f). This defines two 2-functors CatProf that are the identity on objects. The relationship between Cat and Prof encoded in this way makes them into an equipment.

There are also enriched and internal versions of Prof.