Profunctors are categories

Here we use the context of Definition 1. (or, equivalently, of Definition 2.) of page ‘Many Faces of a Profunctor’, that is, we identify a profunctor $F:\ct A^{op}\times\ct B\to \Set$ (according to Definition 3.) with the category $\ct F$ that disjointly contains (isomorphic copies of) $\ct A$ and $\ct B$ and for all objects $a\in Ob\ct A,\ b\in Ob\ct B$ , the set $F(a,b)$ is considered as the homset $\ct F(a\ig b)$ and its elements are called heteromorphisms. Their compositions with arrows of $\ct A$ (from the left) and arrows of $\ct B$ (from the right) are given by the action of the functor $F$ on the arrows.
This setup naturally induces a functor $p:\ct F\to\bf 2$ where $\bf 2$ is the category of two objects $0,1$ and a single nonidentity arrow $0\to 1$ whose preimage at $p$ is exactly the collection of all heteromorphisms in $\ct F$ .
In this setting we write $\ct F:\ct A\ag\ct B$ to express that $\ct F$ is a profunctor from category $\ct A$ to category $\ct B$ .

Definition. Let $\ct F$ and $\ct G$ be profunctors $\ct A\ag\ct B$ . By a profunctor morphism $\teta:\ct F\to\ct G$ we simply mean a functor $\ct F\to\ct G$ such that both of its restrictions, to $\ct A$ and to $\ct B$ , are the identity functors.

Note that such a functor $\teta$ indeed straightly corresponds to a natural transformation $F\to G$ where $F,G:\ct A^{op}\times\ct B\to \Set$ are $F(a,b):=\ct F(a\ig b),\ G(a,b):=\ct G(a\ig b)$ .

Definition. A given profunctor $\ct F:\ct A\ag\ct B$ is said to be generated by a class ${\it F}_0$ of heteromorphisms if every heteromorphism $f$ can be written in the form $f = \alpha\,f_0\,\beta$ for some $\alpha\in \ct A,\ f_0\in {\it F}_0,\ \beta\in\ct B$ .

Definition. Let $\ct F:\ct A\ag\ct B$ and $\ct G:\ct B\ag\ct C$ be profunctors. Their profunctor composition $\ct F\ct G$ is the profunctor $\ct A\ag\ct C$ whose heteromorphisms are formal compositions (i.e. ordered pairs) $(f,g)$ of consecutive heteromorphisms $f\in\ct F$ and $g\in\ct G$ , subject to the (tensor-like) relation that $(f\beta,\,g)$ is considered equal to $(f,\,\beta g)$ for any configuration $a\fl f\to b \fl\beta\to b'\fl g\to c$ .

This is basically the free composition of $\ct F$ and $\ct G$ , and we can actually relax it in the spirit of Definition 2.
For that, let $\bf 3$ denote the preorder category of the three element ordered set $\{0,1,2\}$ , i.e. it has three objects: $0,1,2$ and three nonidentity arrows: ${\bf a}_0: 0\to 1,\ {\bf a}_1:1\to 2$ and their composition $0\to 2$ .

Definition. Let $\ct F:\ct A\ag\ct B$ and $\ct G:\ct B\ag\ct C$ be profunctors with $p:\ct F\to \bf 2$ and $q:\ct G\to\bf 2$ . We call a category $\ct U$ equipped with a functor $t:\ct U\to \bf 3$ , a (realized) ternary composition of $\ct F$ and $\ct G$ , if the preimage of ${\bf a}_0$ at $t$ is $\ct F$ and the preimage of ${\bf a}_1$ at $t$ is $\ct G$ .
(Naturally, this condition can be easily rephrased using pullbacks for expressing the preimages.)

Theorem. For fixed profunctors $\ct F:\ct A\ag\ct B$ and $\ct G:\ct B\ag\ct C$ , there is a correspondence between their ternary compositions and profunctor morphisms of the form $\ct F\ct G\to \ct H$ where $\ct H$ is any profunctor $\ct A\ag\ct C$ .

Proof. For a given ternary composition $t:\ct U\to\bf 3$ we can define $\ct H$ by simply deleting $\ct B$ ( $=t^{-1}(1)$ ) from $\ct U$ and map the formal composition $(f,g)$ of consecutive heteromorphisms $f\in\ct F,\ g\in\ct G$ to their actual composition $fg$ defined in $\ct U$ .
For a given profunctor morphism $\tau:\ct F\ct G\to\ct H$ we can define the category $\ct U$ as basically the union of $\ct F,\ct G$ and $\ct H$ such that the composition of consecutive heteromorphisms $f,g$ is defined to be $(f,g)^\tau$ .

Corollary. We could have defined the composition $\ct F\ct G$ of profunctors $\ct F:\ct A\ag\ct B,\ \ct G:\ct B\ag\ct C$ as the initial object in the category of their ternary compositions.