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 
.