Let $\cal K$ be a 2-category, supporting two Yoneda structures, induced by a pseudoadjunction $P^\sharp\dashv P : {\cal K}^\text{coop}\to \cal K$; this means that there is such an adjunction, or in other words $$ {\cal K}(A, PB)\cong {\cal K}(B,P^\sharp A)^\text{op} \tag{$\star$} $$ naturally in all arguments, and that
- We are given an ideal of $P$-admissible 1-cells and an ideal of $P^\sharp$-admissible 1-cells, and "Yoneda embedding maps" $y_A : A\to PA$ and $y^\sharp_A : A\to P^\sharp A$.
- For every admissible $f : A\to B$ with admissible domain, we are given arrows $B(f,1)$ exhibiting the left extension of $y_A$ along $f$, and $B(1,f)$ exhibiting the right extension of $y_A^\sharp$ along $f$.
These data satisfy axioms YS1, YS2, YS3 that you can read on the $n$Lab page.
I'm trying to prove two things:
- $P$-admissible objects and $P^\sharp$-admissible objects coincide;
- If $f : A \leftrightarrows B : u$ is an adjunction, $B(f,1) : B\to PA$ and $B(1,u) : A\to P^\sharp B$ correspond each other under the adjunction $(\star)$.
In $\bf Cat$, with $P=[\,\_°, Set]$ and $P^\sharp=[\,\_\,, Set]°$, 1. and 2. are true.
