$\require{AMScd}$Some time ago I asked a very specific question about profunctors $$\begin{CD} {\cal C}/A @>\mathfrak p>> {\cal C}/B\end{CD}$$ between slice categories of a fixed category $\cal C$. I haven't made much progress. Let me recap all I know.
Unwinding the definition, a $\mathfrak p$ as above consists of a functor $$\begin{CD} {\cal C}/B @>\mathfrak p>> [({\cal C}/A)^\text{op}, {\sf Set}]\end{CD}$$ which in turn is a functor $$\begin{CD} {\cal C}/B @>\mathfrak p>> [{\cal C}^\text{op}, {\sf Set}]/yA\end{CD}$$ (because the presheaf category over a slice is a slice over the presheaf representable at the object I was slicing), and since a slice category is a comma object this essentially amounts to a functor $p : {\cal C}/B \times {\cal C}^\text{op} \to {\sf Set}$, plus a cocone to $yA$, i.e. a natural transformation $\delta : p \Rightarrow yA$ with components $$\begin{CD} \delta_{h, C} : p(h, C) @>>> {\cal C}(C,A)\end{CD}$$ when $h\in {\cal C}/B$ and $C\in \cal C$. This is a particularly concrete expression for a $\mathfrak p$ like above, and I tried for a while to find examples of such functors, then going back to the profunctor $\mathfrak p$ they are equivalent to. I found some, but they are all representable (in the sense of being induced by a functor between the slices), and I'm pretty sure there are non-representable examples.
So, the question at this point might be summarised as:
Is the above characterization of ${\sf Prof}({\cal C}/A, {\cal C}/B)$ useful in order to build non-representable examples of profunctors $\mathfrak p : {\cal C}/A \to {\cal C}/B$? How can I find some examples of such profunctors?