Is there a Yoneda structure on $\bf PDer$?

Question

The title says it all.

A Yoneda structure in a 2-category, as defined in [SW], is given by a coherent choice of a 1-cell $y_A : A \to PA$ such that

1. $\text{Lan}_{y_A}F\dashv \text{Lan}_F {y_A}$ for each $F :A\to B$;
2. $F(-)\cong \text{Lift}_{B(F-,=)}y_A$;
3. $\text{Lan}_yy\cong 1_{PA}$ (read as: ``the Yoneda embedding is dense'');
4. $\text{Lan}_{y_AF}y\circ \text{Lan}_Gy_A \cong \text{Lan}_{GF}y_A \; (\cong \text{Lan}_G\text{Lan}_F y_A)$.

Of course, the validity of these statements in $\bf CAT$ is a consequence of your favourite way to compute pointwise Kan extensions.

I am wondering if the 2-category of prederivators, i.e. strict functors in $[{\bf Cat}°,{\bf CAT}]$, with pseudonatural transformations and modifications as 1- and 2-cells, has a Yoneda structure.

---

[SW] : Street, Ross, and Robert Walters. "Yoneda structures on 2-categories." Journal of Algebra 50.2 (1978): 350-379, doi: 10.1016/0021-8693(78)90160-6.

Answer 1

This is contained in $\S$6 (Corollary 6.7) of Ross Street's paper

Street, Ross. Conspectus of variable categories. J. Pure Appl. Algebra 21 (1981), no. 3, 307--338, doi: 10.1016/0022-4049(81)90021-9.

where it is shown more generally that each 2-category of the form $\mathrm{Hom}(\mathcal{C}^\mathrm{op},\mathbf{CAT})$, where $\mathcal{C}$ is a small 2-category, has a Yoneda structure.

Note that $\mathrm{Hom}(\mathcal{C}^\mathrm{op},\mathbf{CAT})$ is the 2-category whose objects are the pseudofunctors $\mathcal{C}^\mathrm{op} \to \mathbf{CAT}$, whereas your 2-category of interest is the full sub-2-category of one of these on the strict 2-functors. But as this full sub-2-category is biequivalent to $\mathrm{Hom}(\mathcal{C}^\mathrm{op},\mathbf{CAT})$, the Yoneda structure should transport to one on the full sub-2-category.