| definiendum | ${\bf C}\in{\bf Cat}$ |
| postulate | ${\bf C}$ … category |
| postulate | $\mathrm{Ob}_{\bf C},\mathrm{Mor}_{\bf C} $ … small |
| for all | ${\bf D}\in{\bf Cat}$ |
| postulate | ${\bf Cat}[{\bf C},{\bf D}]$ … functor category $({\bf C},{\bf D})$ |
${\bf Cat}$ is the archetypical example for what is called a 2-cateogry: Each hom-class ${\bf Cat}[{\bf C},{\bf D}]$ is again a (ordinary) category.
Specifically, in ${\bf Cat}$, the hom-classes are functor categories and the hom-classes of those are natural transformations.
| predicate | ${\bf C}$ … small $\equiv {\bf C}$ in ${\bf Cat}$ |
In a small category, both $\mathrm{Ob}_{\bf C}$ and $\mathrm{Mor}_{\bf C}$ are proper sets. See Set universe for the definition of the smallness predicate.
The category of small posets is small itself. But for example, the categories of small sets, small topological spaces, small vector spaces or small groups is not small. The latter are locally small, however.
nLab: Cat, Small category, Large category