TITLE: Ends as a "cotrace" operation on profunctors
QUESTION [8 upvotes]: As mentioned here, there is a trace operation on the monoidal category of profunctors given by taking coends: for any profunctor $F : A\times X \nrightarrow B \times X$, there is a profunctor $Tr^X(F) : A \nrightarrow B$ defined by
$$Tr^X(F)(a,b) = \int^x F((a,x),(b,x))$$
I am interested in the sort of dual operation, where we take an end rather than a coend:
$$CoTr^X(F)(a,b) = \int_x F((a,x),(b,x))$$
Note that $CoTr^X(F)$ has the same type ($A \nrightarrow B$) as $Tr^X(F)$.
My question is,

To what extent can the operation $CoTr^X$ be seen as a "cotrace"?

I've found a bit of information googling on "cotraces", but nothing very comprehensive.  Specifically, I'd like to know the following:

First of all, is there a commonly-accepted definition of "cotrace", and what is its relationship to the trace?
Is there a string-diagrammatic definition?
One way to view the operation $CoTr^X$ is as a limited form of closure for the "external monoidal" structure on profunctors, in the sense that
$${\bf Prof}(G, CoTr^X F) = {\bf Prof}(G \times Hom_X, F)$$
holds naturally in $G : A \nrightarrow B$.  That is, we can view $CoTr^X(F)$ as "$Hom_X \multimap F$".  Is this part of the general definition of cotrace (assuming the answer to (1) is positive), or is it a special feature of this particular operation on Prof?

REPLY [3 votes]: Your exact set of questions appears to have languished unanswered for some time, but I can offer at least a partial answer.
You appear to have rediscovered the notion of Tambara modules.
The comonad $CoTr^X$ is talked about in some depth in Doubles for Monoidal Categories by Pastro and Street. They also dig into the left adjoint of this construction, which is a monad on $Prof$, such that the "strong" profunctors are just its algebras. They talk a fair bit about point #3 as well, though as $Hom$ is the unit for profunctor composition, they can just fuse it away.