TITLE: Theory of promonads
QUESTION [16 upvotes]: I'm led to define a promonad in $\bf D$ as a monoid in the category of endo-profunctors of a category $\bf D$, where the product of two profunctors is their composition as profunctors:
$$
F\odot G := \int^D F(-,D)\times G(D,-)
$$
Is this theory developed in a general fashion in any book/article? 
In the case $\bf D$ is small discrete, promonads in $\bf D$ correspond to small categories having $\bf D$ as set of objects, which I find quite astounding: can this result be generalized to the case where $\bf D$ is any small category, leading to a notion of "thick" category? What's the intuition behind this structure?
And how can I recognize in $\text{Pro-Mnd}(\bf D)$ various categories I can obtain from the set $\bf D$ (the discrete one should correspond to a "trivial" promonad, the maximally connected groupoid to another promonad which I'm not able to characterize)?

REPLY [5 votes]: Before we start some general facts about cocomplete monoidal closed category:

in any category of this kind $\mathcal V$ we can build up the bicategory of monoid, bimodules and bimodules morphisms;
fixed any monoid $M$ in this bicategory we can consider the monoidal category of $M,M$-bimodules;
in such monoidal category a monoid is just a monoid $N$ in $\mathcal V$ with a monoid morphism $M \to N$.

This last fact is the generalization of the well know fact about  $R$-algebras, i.e. that every $R$-algebra can be viewed as an $R,R$-bimodule with a $R$-linear associative and unital multiplication or as a ring with a ring homomorphism from $R$ (we remind that rings are monoid in the cocomplete closed monoidal category $\mathbf {Ab})$.
Now let's do some work!
John Baez told that we can recover the bicategory of sets, spans between set and span morphisms as the bicategory of monoids, bimodules and bimodules morphisms on $\mathbf {Set}^\text{op}$.
A monad in such bicategory is nothing more than a category (as you stated in the question) and a bimodule is just a profunctor.
So the bicategory of categories, profunctors and natural transformation between those is nothing but the bicategory of monoids, bimodules and bimodules morphisms in $\mathbf {Set}^\text{op}$.
Ok now back to your question: what is a monad in such bicategory? Well for start a monad is a monoid in one of the monoidal category of $C,C$-bimodules (a.k.a. profunctors). 
A monad should just be a $C,C$-bimodule for some category/monoid $C$, with a $C$-linear monoidal structure.
From what we have said above this is data are equivalent to a monoid $N$ in the monoidal category of spans in $\mathcal{O}(C)$ (so $N$ is just a category) with a monoid morphism (i.e. a functor) from the category $C$ to $N$.
Hope this helps.