I have recently stumpbled upon a situation that I am trying to describe categorically, and it seems both familiar and not quite exactly what it should be. After idealizing the situation and peeling off the technical complication, it looks like this:
I start with two functors $F : \mathcal{C}^{\mathrm{op}}\to \mathcal{D}$ and $G : \mathcal{C}\to\mathcal{E}$. I can then assemble these two functor into a single bifunctor. $$ \begin{array}{ccrcl} T & : & \mathcal{C}^{\mathrm{op}}\times \mathcal{C} &\to & \mathcal{D}\times\mathcal{E} \\ & & (c,c') & \mapsto &(Fc,Gc') \end{array} $$
It so happens that I know an object $x$ in the category $\mathcal{D}\times \mathcal{E}$, such that for all object $c$ in $\mathcal{C}$, there is a morphism $x\to T(c,c)$ in $\mathcal{D}\times \mathcal{E}$. I believe (I haven't checked it, but it is not really the core of my question) that this association is functorial in both its variables, hence defining a morphism to the end $x\to \int_{c\in\mathcal{C}}T(c,c)$.
Here is where things become a bit weird: for all $c,c'$ in $\mathcal{C}$ I have a function $\mathcal{D}(Fc',Fc) \to \mathcal{E}(Gc, Gc')$. For a morphism $f\in\mathcal{D}(Fc',Fc)$, I will denote $f^* \in\mathcal{E}(Gc,Gc')$, because it reminds me of some kind of adjunction (as with linear operators). I am interested in those morphisms $f\in\mathcal{D}(Fc',Fc)$ such that the following diagram commutes $$ \require{AMScd} \begin{CD} x @>>> T(c',c');\\ @VVV @VV{(f,id)}V \\ T(c,c) @>>{(id,f^*)}> T(c',c); \end{CD} $$
A priori, there is no reason for arbitrary morphisms to satisfy these property, but the ones that do are of interest to me. The way I can understand this condition, is saying that I want $f$ and $f^*$ to really really define adjoint action with respect to $T$. It seems quite a natural sthing to ask, and I would like to further simplify this condition, to see if I can get a nice theoretical view of the situation.
One thing I have tried is to try and get rid of the quantifiers in my assumption, So I can define the two following profunctors
$$ \begin{array}{ccrclccccrcl} P_F & : & \mathcal{C}\times \mathcal{C}^{\mathrm{op}} & \to & \mathbf{Set} & \quad & \quad & P_G & : & \mathcal{C}^{\mathrm{op}}\times \mathcal{C} & \to & \mathbf{Set}\\ & & (c,c') & \mapsto & \mathcal{D}(Fc',Fc) & & & & & (c,c') & \mapsto & \mathcal{E}(Gc,Gc') \end{array} $$ and interprete the operation $f\mapsto f^*$ as some sort of a dinatural transformation between these profunctors (of course one has to check some axioms about it, but I am first assuming in first approximation). But even with that I am not sure how get any further. To be honest, I am not exactly sure what the condition I arrive should even look like, but I find my current condition somewhat clunky and it feels unfinished