I was wondering if there is an easy way to check whether a given profunctor $$ p : {\cal C}^{op}\times{\cal D} \to \sf Set $$ is "representable" in the sense that $p$ is the functor ${\cal D}(F,1)$ for some functor $F :{\cal C} \to {\cal D}$.
More precisely, I was wondering if it is possible to read representability in the "collage" of $p$, i.e. its category of elements, meaning that I would like something that sounds as
the following conditions are equivalent for $p : {\cal C}^{op}\times{\cal D} \to \sf Set$:
- $p$ is representable;
- $El(p)$ has property $\Gamma$.
for some property $\Gamma$ to be determined.
This condition can certainly not be that $El(p)$ has a terminal object, because even if $p$ can certainly be regarded as a presheaf on ${\cal C}\times {\cal D}^{op}$, that condition yields something different/stronger, i.e. the fact that $$ p(X,Y) = {\cal C}(X,U) \times {\cal D}(V,Y)$$ for a unique object $(U,V)\in {\cal C}\times{\cal D}^{op}$ (and consequently $p$ acts on morphisms as pre/post composition). So, there's also a small clash of notation, because a profunctor $p$ is not representable in the sense above iff it is representable as a presheaf on ${\cal C}\times{\cal D}^{op}$... which is fine, but mixing the two terminologies doesn't allow me to exploit "representability" theorems for presheaves as black boxes.
Is there something obvious I fail to see?
I suppose by $\mathcal D(F,1)$ you mean the functor $(c,d)\mapsto \mathcal D(Fc,d)$.
Then $\Gamma$ is simply:
We obtain $F$ (up to natural isomorphism) as the reflection functor (restricted to $\mathcal C$).
Note that the category of elements is a bit different than the collage category: it has the heteromorphisms (elements of $p(c,d)$) as objects.
Actually it's the full subcategory of a 'flipped' arrow category of the collage, restricted to heteromorphisms.