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 (according to Definition 3.) with the category that disjointly contains (isomorphic copies of) and and for all objects , the set is considered as the *homset* and its elements are called *heteromorphisms*. Their compositions with arrows of (from the left) and arrows of (from the right) are given by the action of the functor on the arrows.

This setup naturally induces a functor where is the category of two objects and a single nonidentity arrow whose preimage at is exactly the collection of all heteromorphisms in .

In this setting we write to express that is a profunctor from category to category .

**Definition.** Let and be profunctors . By a *profunctor morphism* we simply mean a functor such that both of its restrictions, to and to , are the identity functors.

Note that such a functor indeed straightly corresponds to a natural transformation where are .

**Definition.** A given profunctor is said to be *generated* by a class of heteromorphisms if every heteromorphism can be written in the form for some .

**Definition.** Let and be profunctors. Their *profunctor composition* is the profunctor whose heteromorphisms are formal compositions (i.e. ordered pairs) of consecutive heteromorphisms and , subject to the (tensor-like) relation that is considered equal to for any configuration .

This is basically the *free* composition of and , and we can actually relax it in the spirit of Definition 2.

For that, let denote the preorder category of the three element ordered set , i.e. it has three objects: and three nonidentity arrows: and their composition .

**Definition.** Let and be profunctors with and . We call a category equipped with a functor , a *(realized) ternary composition* of and , if the preimage of at is and the preimage of at is .

(Naturally, this condition can be easily rephrased using pullbacks for expressing the preimages.)

**Theorem.** For fixed profunctors and , there is a correspondence between their ternary compositions and profunctor morphisms of the form where is any profunctor .

** Proof.** For a given ternary composition we can define by simply deleting () from and map the formal composition of consecutive heteromorphisms to their actual composition defined in .

For a given profunctor morphism we can define the category as basically the union of and such that the composition of consecutive heteromorphisms is defined to be .

**Corollary.** We could have defined the composition of profunctors as the *initial object* in the category of their ternary compositions.