Speaking of companions and conjoints, I was wondering whether the cap and cup profunctors can be written as companions or conjoints of a functor.
I think that \\(\text{hom} : \mathcal{X}^{\text{op}} \otimes \mathcal{X} \to \mathcal{V}\\) is the companion of the identity \\(\mathcal{V}\\)-functor \\(\mathbf{1} : \mathcal{X} \to \mathcal{X}\\).
So I was kind of expecting to get something similar for cups and caps, but I didn't manage to.