Locally Small Category construction confusion:

Question

Working through Categories for the Working Mathematician, there was a point that has confused me. Throughout chapter 2, section 2, Mac Lane describes a (family of) functors which map locally small categories to Set by $b \mapsto \hom(a, b)$ and $f \mapsto f_*$. At the top of page 35 he then describes a method for constructing a similar functor on categories that aren't locally small:

To include categories [that aren't locally small], we can proceed as follows: Given a category $C$, take a set $V$ large enough to include all subsets of the set of arrows of $C$...

I'm confused by this: how could $V$ possibly be a set and not a proper class? Indeed, how could all the arrows of $C$ not be a proper class? Since $C$ is not locally small, there is some $a, b$ such that $\hom(a, b)$ is a proper class. Then, $\hom(a, b)$ is a subset of all arrows of $C$, which in turn has an obvious bijection to $V$, by sending a given arrow to its singleton. Thus, $\hom(a, b) \subseteq V$, so $V$ must also be a proper class. What am I missing?

Answer 1

Mac Lane defines "small sets" in terms of universes. In particular, on page 12 he says:

We shall assume that there is a big enough set $U$, the "universe", then describe a set $x$ as "small" if it is a member of the universe.

In section I.6, he elaborates that by "big enough set" he actually means a Grothendieck universe. So for him, a "large" category is still a set, just one which is larger than the chosen universe. For categories that are actually proper classes, he uses the term "metacategory" instead (and does not consider them to be "categories").