JesÃºs and Matthew, thank you very much for your comments, which I need more time to absorb.

But for now I would add that it seems that we can at least argue for defining a category like \\(Cat\\) in two very different ways. That's because it depends on what we mean by Object. In the case of a set, then we typically suppose that the objects exist independently of, and prior to, the set. Sets are defined bottom up, and there can be different sets based on different objects. {1,2,3} is different from {a,b,c}.

But the whole point of Category theory, as much as I imagine, is that you can define things top down. They exist only to the extent that YOU define them. And you define them by structuring them. So they must be defined by their relationships with themselves. And so from the Category theory point of view, (or let us say, the Structure point of view), you don't define a single set, but rather, you define ALL sets at once. You define "Set-ness". And "Set-ness" is defined by set-functions. But my understanding is that it's not how set-functions works (as maps between sets) that's relevant. Instead, it is the structure of all of the maps that is relevant. If you had another category which had the same relationships (though with nominally different objects), then it wouldn't just be isomorphic, it would be EQUAL to the category of sets. Because it's an equality of CATEGORIES that matters, not an equality of objects. Similarly, 3 apples is different from 3 oranges, but the two 3s are equal! 3 apples is a different set than 3 oranges, but the category 3-ness is the same.

Maybe I'm wrong. But at least you see how I'm thinking.