$\def\L{\mathfrak{L}}\def\Prof{\mathsf{Prof}}\require{AMScd}$ The notation for this question is the same of this post: in particular
Isbell duality $\text{Spec}\dashv {\cal O} : {\cal V}^{A^°} \leftrightarrows \big({\cal V}^A\big)^°$ allows us to define the functor $$ \L : \Prof(A,B) \to \Prof(B,A) $$ ($\cal V$ is a cosmos in which $A,B$ are enriched categories) sending $K : A^° \times B \to \cal V$ into $\L(K) : (b,a)\mapsto {\cal V}^{A^°}(K(-,b), \hom(-,a)) = {\cal O}(K_b)_a$.
- If $Q : A' \to A$ is a profunctor, there is a diagram $$ \begin{CD} \Prof(A,B) @>\L>> \Prof(B,A) \\ @VQ^*VV @AAQ_*A\\ \Prof(A',B) @>>\L> \Prof(B, A') \end{CD} $$ of course, since $Q$ is arbitrary, there is no hope that this commutes; but there is a canonical 2-cell filling it?
- It is rather easy to find that there is a 2-cell $K\diamond \L(K) \Rightarrow \hom_B$ (let's call it $\epsilon$ since it behaves like an "evaluation").
More in detail, we can consider the natural transformation $\epsilon : K \diamond \L(K) \Rightarrow \hom_B$ for a given $K$, obtained from the mate $$ K(b',a)\otimes \textsf{Nat}(K(-,a), y_b) \to \hom(b',b) $$ of the projection $$ \textsf{Nat}(K(-,a), y_b) \to {\cal V}(K(b',a),\hom(b',b)). $$ Since this is a cowedge, $\epsilon$ is now obtained as the map $P\diamond \L(P)(b',b)=\int^a K(b',a)\otimes \textsf{Nat}(K(-,a), y_b) \to\hom(b',b)$.
Nevertheless, $K$ is rarely a left adjoint for $\L(K)$ (a sufficient condition is that $K$ is representable, so that $K=\hom(k,1), \L K = \hom(1,K)$). Is this condition also necessary? Does $\epsilon$ satisfy a universal property?
