ct.category theory - 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 $Xicolon mathrm{Cat}to
mathrm{Cat}$. I want to consider a certain "intertwining" functor
$$
Vcolon Ximathrm{PSh}(C) to mathrm{PSh}(Xi C)
$$
defined by the formula
$$
(VX)(gamma) = mathrm{Hom}_{Ximathrm{Psh}(C)}(Agamma, X),
$$
where $X$ is an object of $Ximathrm{PSh}(C)$, $gamma$ is an object
of $Xi C$, and $Acolon Xi Cto
Ximathrm{PSh}(C)$ is the functor obtained by
applying $Xi$ to the Yoneda functor $Cto mathrm{PSh}(C)$.

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 $Xi C= Ctimes C$. Then $Vcolon mathrm{PSh}(C)times
  mathrm{PSh}(C)to mathrm{PSh}(Ctimes C)$ 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 $Vcolon
  mathrm{PSh}(C)^{mathrm{op}} to mathrm{PSh}(C^{mathrm{op}})$ is
  a sort of "dualizing" functor, which sends a presheaf $X$ on $C$
  to the presheaf $cmapsto mathrm{Hom}_{mathrm{PSh}(C)}(X, Fc)$ on
  $C^mathrm{op}$; here $Fcolon Cto mathrm{PSh}(C)$ represents the
  Yoneda functor.




• Let $Xi C=mathrm{gpd} C$, the maximal subgroupoid of $C$.
  Then $Vcolon mathrm{gpd}\,mathrm{PSh}(C)to
  mathrm{PSh}(mathrm{gpd}C)$ 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.

1. What makes a functor $Xi$ reasonable? Is it enough if it's
   accessible?




2. I think $V$ should be the left Kan extension of the Yoneda
   functor $Bcolon Xi Cto mathrm{PSh}(Xi C)$ along $A$. Is this
   true? When can I expect to have $VAapprox B$?




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




4. Given a functor $fcolon Cto D$, 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.