/sci/ - Science & Math

>>12192067
>since that's not the case for a identity natural transformation of a single functor that sends every morphism to a single one, for instance.
I'm too tired already to think about whether I'm wrong about all possible arrows in a square having to pass through the same two arrows of the natural transformation.

But be that as it may, I don't recall people demanding both doing ZF and also working with categories that are classes. If you want to work in ZF, then e.g. instead of saying "the category of groups", I'm sure you can get far by saying I you consider a set (not specifying that set, leaving room for it to be a large cardinal too, should you later choose to postulate it) where all elements (already in it) are groups. I'm confident that whatever closedness properties of that set you actually need, ZF will do.

For a more educated take, there's that paper by Shulman
https://arxiv.org/pdf/0810.1279.pdf