Is $Cat$, the category of all the small categories, a small category?

Question

Does $Cat$, the category of all the small categories, contain $Cat$ itself? I.e. is $Cat$ a small category?

The following quote from Categories from the Working Mathematicians says $\times : Cat \times Cat \to Cat$ is a functor. Does it mean that the domain $Cat \times Cat$ is a (small) category, and $Cat$ is a (small) category?

The product $\times$ is thus a pair of functions: To each pair $(B, C)$ of categories, a new category $B \times C$; to each pair of functors $(U, V)$, a new functor $U \times V$. Moreover, when the composites $U' U$ and $V' V$ are defined, one clearly has $(U' \times V') (U \times V) = U' U \times V'V$. Hence the operation $\times$ itself is a functor; more exactly, on restricting to small categories, it is a functor $\times : Cat \times Cat \to Cat$.

Thanks.

Answer 1

No, elements of $Cat$ are small categories - $Cat\times Cat$ is the domain, and $Cat$ the codomain, of $\times$. Analogously, "a function on natural numbers" would be $f:\mathbb{N}\rightarrow\mathbb{N}$ but that wouldn't imply that $\mathbb{N}$ is a natural number.

Indeed, $Cat$ itself is not a small category, since there are a proper class of small categories (easy exercise: there is a category of cardinality $\kappa$ for every cardinal $\kappa$).

(Of course, $Cat$ is locally small - given any two small categories there are only set-many functors between them - but it's not small.)