I am currently in the process of changing the way I think about category theory, by adopting the notion of Grothendiek universe and trying not to think of proper classes.
Think of a statement such as
If $\mathcal C$ and $\mathcal D$ are categories, and $F,G \colon \mathcal C \to \mathcal D$ are functors, then $ \int_{c \colon \mathcal C} \mathcal D(Fc, Gc) = \mathrm{Nat}(F,G). $
This doesn't make much sense as written, since depending on the foundations you use there are potentially size problems and $\mathsf{Set}$ might not recive the functor $\mathcal D(F-,G-)$.
Suppose that we are using MacLane's foundations of set theory, so that $\mathrm{Nat}(F,G)$ is a set, and the objects of a category form a set, as do all the arrows. Let $\mathrm{Set}[U]$ be the category of sets whose object set is the universe $U$. Is the following an appropriate formalisation of the above statement?
Let $\mathcal C$ and $\mathcal D$ be categories, and let $F,G \colon \mathcal C \to \mathcal D$ be functors. Let $U$ be a Grothendiek universe such that $\mathsf{Set}[U]$ receives each functor $\mathcal D (F-, G-)$ and $U$ contains $\mathrm{Nat}(F,G)$. Then $\mathrm{Nat} (F,G)$ is the end of $\mathcal{D}(F-,G-).$
Have I got the right end of the stick? Is that even true?
Does $U$ necessarily have to be a universe for the above to be true?
Also if given a universe $U$, and a bigger one $U'$, is it true that the inclusion $\mathsf{Set}[U] \hookrightarrow \mathsf{Set}[U']$ will always preserve limits? Would it at least always preserve ends?