I've been reading some category theory texts for my undergraduate monography, and I've found that one can talk about small and large categories using small/large sets, or do the same using sets/classes. Most introductory texts just shrug formalisms off, mention one of these formalizations and goes on.
I've noticed that most category-focused authors and texts (such as Categories for the working mathematician and Sheaves in Geometry and Logic by Mac Lane) tend to prefer the small/large set approach, while texts that don't focus that much on categories (Topology and Groupoids by R. Brown) tend to prefer the class/set one.
My question:
Is there a categorical/set-theoretical reason to prefer the small sets approach?
I'd assume that there's a benefit in having a set of objects even in large categories, but fail to see an actual difference. I found the paper Set Theory for Category Theory by Michael Shulman, but it was way out of the scope of what I can understand.