The categories Set and Ens

I’ve recently started reading Categories for the Working Mathematician and I’m a little puzzled about the distinction between $\textbf{Set}$ and $\textbf{Ens}$. On the one hand, it seems like $\textbf{Ens}$ is supposed to be a full subcategory of $\textbf{Set}$:

If $V$ is any set of sets, we take $\textbf{Ens}_V$ to be the category with objects all sets $X \in V$, arrows all functions $f : X \to Y$, with the usual composition of functions. By $\textbf{Ens}$ we mean any one of these categories.

And on the other hand it seems like $\textbf{Ens}$ can be bigger than $\textbf{Set}$:

$D$ must have small hom-sets if these functors are to be defined […]. For larger $D$, the Yoneda lemmas remain valid if $\textbf{Set}$ is replaced by any category $\textbf{Ens}$ […] provided of course that $D$ has hom-sets which are objects in $\textbf{Ens}$.

I suspect my confusion is arising from a misinterpretation of what $\textbf{Set}$ is and what it is used for. Mac Lane seems to make a point of building $\textbf{Set}$ using a fixed universal set $U$, but is there any benefit to doing this instead of simply taking $\textbf{Set}$ to be his “metacategory” of all sets? Indeed, set-theoretically, can such a universal set exist? Mac Lane requires the following properties:

  1. $x \in u \in U$ implies $x \in U$,
  2. $u \in U$ and $v \in U$ implies $\{ u, v \}$, $\langle u, v \rangle$, and $u \times v \in U$.
  3. $x \in U$ implies $\mathscr{P} x \in U$ and $\bigcup x \in U$,
  4. $\omega \in U$ (here $\omega = \{ 0, 1, 2, \ldots \}$ is the set of all finite ordinals),
  5. if $f : a \to b$ is a surjective function with $a \in U$ and $b \subset U$, then $b \in U$.

As far as I can tell, this is essentially a transitive inner set-model, but isn’t the existence of such a thing unprovable in ZFC? I haven’t gotten very far through the book yet, but it seems to me that there is no loss in reading “small set” as “any set” and non-small sets as proper classes…

Solutions Collecting From Web of "The categories Set and Ens"

In Categories for the working mathematician, Mac Lane assumes the existence of one Grothendieck universe $U$, and $\mathbf{Set}$ is the category of sets in $U$. This device ensures the existence of functor categories like $[\mathbf{Set}, \mathbf{Set}]$.

On the other hand, $\mathbf{Ens}$ is any full subcategory of the metacategory of all sets, with the restriction that the collection of objects in $\mathbf{Ens}$ is itself a set. Note that $\mathbf{Ens}$ may fail to have the properties expected of $\mathbf{Set}$, e.g. cocompleteness, cartesian closedness, etc.