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.




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[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](https://doi.org/10.1016/0021-8693(78)90160-6).



Is there a Yoneda structure on $\bf PDer$?
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](https://doi.org/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.


