TITLE: Is there lore about how endofunctors of Cat interact with the formation of presheaf categories?
QUESTION [11 upvotes]: 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.

REPLY [12 votes]: 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.