A category which is not small may be called large, especially when it is not essentially small (see below).

Properties

Small categories are free of some of the subtleties that apply to large categories.

A category is said to be essentially small if it is equivalent to a small category. Assuming the axiom of choice, this is the same as saying that it has a small skeleton, or equivalently that it is locally small and has a small number of isomorphism classes of objects.

A small category structure on a locally small category$C$ is an essentially surjective functor from a set (as a discrete category) to $C$. A category is essentially small iff it is locally small and has a small category structure; unlike the previous paragraph, this result does not require the axiom of choice.

Smallness in the context of universes

If Grothendieck universes are being used, then for $U$ a fixed Grothendieck universe, a category $C$ is $U$-small if its collection of objects and collection of morphisms are both elements of $U$. $C$ is essentially $U$-small if there is a bijection from its set of morphisms to an element of $U$ (the same for the set of objects follows); this condition is non-evil.

So let $U\Set$ be the category of $U$-small sets. Then

a $U$-small category is a category internal to$U Set$.

A category is $U$-moderate if its set of objects and set of morphisms are both subsets of $U$. However, some categories (such as the category of $U$-moderate categories!) are larger yet.