Let U be a Grothendieck universe, and U<sup>+</sup> its successor universe (assume Grothendieck's universe axiom).

Everybody agrees that a U-small category is a category whose sets of objects and morphisms are both elements of U.  For the "next larger size" of categories which are not necessarily even locally small, call them just U-categories, there are two possible definitions:

* a category whose sets of objects and morphisms are both subsets of U;
* a category whose sets of objects and morphisms are both elements of U<sup>+</sup> (U<sup>+</sup>-small categories).

I quite prefer the second notion, so that the category of U-categories is cartesian closed and we can form localizations.  This is the usage of Dwyer-Hirschhorn-Kan-Smith, "Homotopy Limit Functors on Model Categories and Homotopical Categories".  I think the first more closely corresponds to non-Grothendieck universe-based treatments of category theory using sets and classes, but I might be wrong about that.

For U-locally small categories there are again two possible definitions:

* a category whose set of objects is a subset of U and whose set of morphisms is an element of U,
* a category whose set of objects is an element of U<sup>+</sup> and whose set of morphisms is an element of U.

I don't see a strong reason to prefer one over the other, except that the second is more parallel with my preference for U-categories.  DHKS uses the first.  As an example of the difference between them, if I have a U-locally small category C, I can form the category (poset) of full subcategories of C; this is U-locally small under the second definition, but not the first.  Is this a good thing or a bad thing?  Or are there no theorems I would care about that are affected by this difference?  Does anyone have an opinion about these two definitions?