nLab
Prof

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

Recall that a profunctor from $A$ to $B$ is a functor $B^{\mathrm{op}}\times A\to \mathrm{Set}$. Composition of profunctors in $\mathrm{Prof}$ is by the “tensor product of functors” coend construction: if $H:A\to B$ and $K:B\to C$, their composite is given as a functor $C^{\mathrm{op}}\times A\to \mathrm{Set}$ by

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

The identity on a category $A$ is its hom-functor ${\mathrm{Hom}}_A(-,-)$.

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

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

There are also enriched and internal versions of $\mathrm{Prof}$.