The way category theory was introduced to me is by defining categories as a class of objects together with a class of morphisms between any two objects, with an asociative composition map and identity morphisms for each object.
The reason given for using classes is that "for set theoretic reasons they cannot be sets". But since classes are also well defined objects in some set theories, where in particular proper classes exist and cannot be contained in any object, this seems a bit restrictive on the possible applications of category theory.
Is it not more useful to leave what the collections of objects and morphisms are purposefully vague, such that everything with categorical behaviour can be classified as a category? In particular the category with all classes as objects and functional classes as morphisms obviously behaves categorically but is not, under the set theoretic definition, a category.
As a secondary question, while it makes sense to link a locally small categories together through the category of sets with representations and Hom-set adjunctions, it again feels restrictive to require a bijective map to a specific category. Is it possible to define representability and Hom-[some different category] adjunction in a way that works for all categories, instead of just locally small categories?
More generally: which properties of locally small categories can be generalised to all categories by omitting the specific choice of Set as a category?
