Eilenberg and Mac Lane formally defined categories in their 1945 paper *General Theory of Natural Equivalences*. Their definition of a category starts as follows:

"*A category {A,a} is an aggregate of abstract elements A (for example, groups), called the objects of the category and etc.*"

When they consider their first example of categories on P. 239, namely the category of all sets, they immediately remark: "*This category obviously leads to paradoxes of set theory.*"

Obviously, all sets collected together don't form a set. However, their definition does not require the objects of a category to form a set. They use the term "aggregate" in their definition, perhaps to avoid this particular issue. That is, a set can be an aggregate, a class (in the sense of NBG) can also be an aggregate. Since the "aggregate" of all sets is a class, I do not see the "paradox" that they say will arise from considering the category of all sets. Unless by "aggregate" they really meant set. **So my question is: what is the paradox that they were referring to?**

Please note, my question is specific to Eilenberg and Mac Lane's comment about the category of all sets. Obviously, there are *other* paradoxes caused by their definition of categories, and allowing the notion of a class doesn't eliminate all set-theoretic paradoxes from category theory. But I am not asking about these topics.

On the other hand, as pointed out on page 245 of R. Kromer's book (Tool and Object): "Eilenberg and Mac Lane use the term 'set' in the combination *the set of all objects of [a] category*, on page 238 of their paper."

**Is this an evidence that by "aggregate" they really meant "set"?**