Is there lore about how endofunctors of Cat interact with the formation of presheaf categories? This is a request for references about a peculiar categorical
construction I've run into in some work I've been doing, and about which I'd
like to learn as much as I can.
Let $\mathrm{Cat}$ be the category of small categories, and let
$\mathrm{PSh}(C)$ be the category of presheaves of sets on a category $C$.
Suppose we
are given a "reasonable" endofunctor $\Xi\colon \mathrm{Cat}\to
\mathrm{Cat}$.  I want to consider a certain "intertwining" functor
$$
V\colon \Xi\mathrm{PSh}(C) \to \mathrm{PSh}(\Xi C)
$$
defined by the formula
$$
(VX)(\gamma) = \mathrm{Hom}_{\Xi\mathrm{Psh}(C)}(A\gamma, X),
$$
where $X$ is an object of $\Xi\mathrm{PSh}(C)$, $\gamma$ is an object
of $\Xi C$, and $A\colon \Xi C\to 
\Xi\mathrm{PSh}(C)$ is the functor obtained by 
applying $\Xi$ to the Yoneda functor $C\to \mathrm{PSh}(C)$.
Note: it's unreasonable to expect for a randomly chosen $\Xi$ that the
category $\Xi \mathrm{PSh}(C)$ is even defined, since
$\mathrm{PSh}(C)$ is a large 
category, and $\Xi$ is given as a functor on small categories.  And even if it is defined, it's unreasonable to expect that $V$ is
well-defined, since $(VX)(\gamma)$ may not be a set.  But here are
some reasonable examples:


*

*Let $\Xi C= C\times C$.  Then $V\colon \mathrm{PSh}(C)\times
  \mathrm{PSh}(C)\to \mathrm{PSh}(C\times C)$ is the "external
product" functor, which takes a pair of presheaves $(X_1,X_2)$ on $C$ to
the presheaf $(c_1,c_2) \mapsto X_1(c_1)\times X_2(c_2)$ on $C^2$.
You can generalize this by considering $\Xi C= \mathrm{Func}(S,C)$,
where $S$ is a fixed small category.

*Let $\Xi C = C^{\mathrm{op}}$.  Then $V\colon
  \mathrm{PSh}(C)^{\mathrm{op}} \to \mathrm{PSh}(C^{\mathrm{op}})$ is
a sort of "dualizing" functor, which sends a presheaf $X$ on $C$
to the presheaf $c\mapsto \mathrm{Hom}_{\mathrm{PSh}(C)}(X, Fc)$ on
$C^\mathrm{op}$; here $F\colon C\to \mathrm{PSh}(C)$ represents the
Yoneda functor.

*Let $\Xi C=\mathrm{gpd} C$, the maximal subgroupoid of $C$.
Then $V\colon \mathrm{gpd}\\,\mathrm{PSh}(C)\to
  \mathrm{PSh}(\mathrm{gpd}C)$ is such that $(VX)(c)$ is the set of
isomorphisms between $X$ and the presheaf represented by $c$.
The sorts of questions I have include the following.


*

*What makes a functor $\Xi$ reasonable?  Is it enough if it's
accessible?

*I think $V$ should be the left Kan extension of the Yoneda
functor $B\colon \Xi C\to \mathrm{PSh}(\Xi C)$ along $A$.  Is this
true?  When can I expect  to have $VA\approx B$?

*How does $V$ of a composite $\Xi \Psi$ relate to the composite of the
$V$s of each term?

*Given a functor $f\colon C\to D$, you get a bunch of functors
between the associated presheaf categories.  How does $V$ interact
with such functors?
There's really only one or two examples of $\Xi$ that really I need to
understand this for, and I don't want to spend time working out a general theory of this thing.  It would be most convenient if someone can
point me to a reference which talks about this construction.  Even one
that deals with particular instances of it would be helpful.
 A: There are a few things going on here. In short, what you are describing is a pseudodistributive law between a lax-idempotent relative pseudomonad and a pseudofunctor.
The presheaf construction, as you mention, must be treated carefully to deal with size issues. One can consider instead the small-presheaf construction $\mathbf{PSH}$, which is a pseudomonad on $\mathbf{CAT}$, the 2-category of locally-small categories, but in many cases we really are interested in presheaves on small categories. The solution is to precompose the small-presheaf pseudomonad by the inclusion $\mathbf{Cat} \hookrightarrow \mathbf{CAT}$ of small categories into locally-small categories, which produces a $(\mathbf{Cat} \hookrightarrow \mathbf{CAT})$-relative pseudomonad $\mathbf{Psh}$. This is discussed in some detail in the paper Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures of Fiore–Gambino–Hyland–Winskel.
Just as there is a notion of pseudodistributive law between pseudomonads, so too there is a notion of pseudodistributive law between a relative pseudomonad and a pseudofunctor/pseudomonad. In your case, $\Xi$ is a pseudofunctor on $\mathbf{CAT}$ which restricts to a pseudofunctor on $\mathbf{Cat}$ (cf. §6 ibid.).
A special property of the presheaf construction in particular is that it is lax-idempotent (essentially because it is a cocompletion). This entails that any pseudodistributive law $V : \Xi \mathbf{Psh} \to \mathbf{Psh} \Xi$ must be given by a left extension (with invertible unit), and hence pseudodistributive laws over the presheaf construction are essentially unique (in the case of relative pseudomonads, this does not yet appear in the literature, but see Definition 33, Theorem 35, and Corollary 49 of Walker's Distributive laws via admissibility for the non-relative case).
Therefore, we see that (1) $\Xi$ being reasonable amounts to the existence of nice left extensions, and (2) always holds. (3) When $\Xi$ is a composite, the theory of iterated distributive laws likely holds some answers, with simplifications available in the lax-idempotent setting. (4) Since the composite relative pseudomonad will again be lax-idempotent (Corollary 50 of Walker), one can apply the theory of admissibility for lax-idempotent pseudomonads to obtain necessary and sufficient conditions for admissibility of the composite relative pseudomonad (which is what you call commutativity with the restriction functors).
There are several details to be checked in your specific setting (i.e. pseudofunctors instead of pseudomonads, and relative pseudomonads instead of pseudomonads), but hopefully the situation is clear enough that this should simply involve some routine calculations.
A: This is really just a comment, but it's too long to fit. 
Many people have come up against the problem that PSh isn't an endofunctor of Cat, because even if C is small, PSh(C) usually isn't.  There's a standard way to solve this problem, as follows.  


*

*Replace Cat (small categories) with CAT (locally small categories)

*Replace PSh (presheaves) with psh (small presheaves, i.e. small colimits of representables)


Then psh is genuinely an endofunctor of CAT.  If C is small then psh(C) = PSh(C).  But if C is not small then psh(C) is a proper subcategory of PSh(C).
In fact, psh is not only an endofunctor of CAT, but a monad.  It's free small-cocompletion.  That is, it takes a category and freely adjoins colimits.
The unit of this monad is the Yoneda embedding.  Given this, and given that the Yoneda embedding plays a part in your considerations, I wonder whether the multiplication of the monad plays a part too.
