This is one main reason why the larger mathematical community does not closely follow these foundational discussions (there are foundations that are strict extensions of ZFC but these extensions are usually conservative).
The usual response is that it would be nice if the formal foundations of mathematics corresponded more closely to the informal intuitions of mathematicians, which is arguably not true of set theoretic foundations and perhaps is more true of category theory. Whether or not you agree about the correspondence or whether even if you agree you find it at all convincing is up to you.
The usual response is that it would be nice if the formal foundations of mathematics corresponded more closely to the informal intuitions of mathematicians, which is arguably not true of set theoretic foundations and perhaps is more true of category theory. Whether or not you agree about the correspondence or whether even if you agree you find it at all convincing is up to you.