Yeah but your comment didn't use the concept of a "category" at all, whereas you used "set" three times. Its definitely more natural to say these definitions come from set theory.

To me, "disjoint union", "direct product" and "set of maps" (as a special case of "internal hom") are the central keywords in what I wrote: Concepts from category theory.

Those keywords all appeared in the context of set theory long before category theory was thought of.

Even ignoring that as a historical accident (we invented set theory before category theory), those ideas are all natural, meaningful and interesting to examine in the context of sets even if one has never heard of a category.

Yes, there's no need to invoke any category theory concepts here. Everything is happening in the category of sets.

First, that's not what your link says. (See, e.g., the comments to the first answer.)

Second, I find that claim highly dubious. To do metamathematics and talk about the relative strength of various axiomatic systems, you need to talk about large cardinals. So that, at least to me, seems like a good argument that set theory is foundational to mathematics in a way that category theory is not, since the latter has no non-eliminable place in the (study of) contemporary foundations of mathematics.

But talking about category theory makes you sound smarter.

I'm personally more scared of set theory.