Existence of not locally small categories I had a strange remark answered to one of my questions some time ago. My question was involving "locally small categories", and that comment was saying that the existence of not locally small categories is doubtful. However, I can find (well-defined) examples of such categories on the net, so that it seems clear to me that they exist...
But somebody could explain me what are the origin of the doubts?
 A: There are a number of authors who reserve the word ‘category’ for what others would call locally small categories, e.g. Adámek and Rosický in Locally presentable and accessible categories. (This becomes troublesome later when they start talking about the category of functors between two not-necessarily-small categories!) The issue is purely terminological/philosophical.
Some (other) authors also refer to possibly-class-sized models of the first-order theory of categories as ‘metacategories’, e.g. Mac Lane in Categories for the working mathematician. (Note that in CWM, every category has a set of objects and a set of morphisms, but that Mac Lane distinguishes between small sets and general sets!)
A: The smallness notion is relative to the framework you are working in : to say that something is a set, you have to define what is a set.
Considering a model $\mathcal U$ of a set theory (say ZFC), you can then define the $\mathcal U$-locally small categories. Then you have a canonical example of such a category : the category of $\mathcal U$-small collections, that is the elements of the model $\mathcal U$, and the applications between them.
The doubts are therefore about the existence of such a model $\mathcal U$, i.e. the consistency of ZFC. (See Gödel's second incompleteness theorem.)
