Equality of functors is dangerous. While it technically is possible to ask whether two functors are equal in foundations like ZFC Set Theory, this might violate the principle of equivalence, and thus should be avoided. But functors are supposed to be the maps between categories, and equivalence of categories is often defined in a way that depends on functors. This, i feel might be a source of a lot of bad intuition for newcomers to category theory like me, who might be misled into thinking of categories like algebraic structures, when actually their maps are in a sense weaker (for example, it doesn't really make sense to ask about the cardinality of a category since equivalent categories can have different cardinality).
One alternative way to define equivalence of categories in ZFC, without using functors is by using skeletons. Unfortunately this requires choice. What are some other ways to define equivalence of categories in more general settings (like intuitionistic set theory) without using the notion of a functor?