Suppose I want to establish a theory of the category $C$ (vector spaces or whatever), but what I really have is $D$, some precisely known category. This is to say, I know all the axioms of $D$, but I only have an intuition for $C$ and I want to develop a theory of $C$. I would normally have diagrams in $C$ by mapping cpos or Domains into $C$. But instead, I want to do it with $D$. What I do is define the largest category, $J$, of Domains in $D$. I do this by defining a dcpo with objects as elements in $D$ and relations as arrows in $D$. Then any functor will map the domains in $J$ to diagrams in $C$.
It seems like I am just inserting a category $D$ in the normal diagram functor $J \rightarrow C$ resulting in $J \rightarrow D \rightarrow C$ which seems to miss the point of the exercise. The point of the exercise, I think, is to try to do a lot of category theory when you have to live in some category $D$.
We start by saying that we "have access to" all diagrams in $D$. Further, we say that we have access to none of the morphisms in any other category. So if we want to talk about a category $C$, it will have to be in terms of diagrams in $D$. Next, we intuit the existence of a category $C$ (I am using this restricted language to reflect the notion that we do don't have access to $C$). Next, we consider endofuntors of $D$, but we really see them as diagrams in $D$ indexed by the domains we constructed in $D$ by $J$. These endofunctors are meant to mimic functors from $D$ to $C$. We are pretending to have access to $C$, by attempting a construction of $C$ in $D$.
Sorry that this is so unclear, especially the idea of having an "intuition of C" and "attempting a construction of". I think that this is an expression of a Topos, and so I have some questions. Firstly, what kind of minimum structure do we need in $D$ to really start doing some work? Second, if we really want to say that we only have access to $D$, then we cannot present $D$ as a set of morphisms and a set of objects because that would imply we are actually in SET, not $D$. Is there any way to start working only in $D$? This goes back to the first question (although thinking about this too much is a bit of a morass).