How to properly defined a category of functors?

Question

I’ve got a foundational question about category theory today :

How do we define a category of functors (and maybe for that matter how do you define a category) ?

I’ve got a problem with the definitions I’ve seen because the categories of functors where define even for non small categories. The problem here is that the morphisms between functors are natural transformations, which are, in non small categories, proper classes. This isn’t allowed in the definition of a category since the morphisms between two objects are supposed to form a set (thus the morphisms need to be sets as they are elements of a set).

How do we resolve this ?

Thanks in advance, Max

Answer 1

This is mainly a fake question and you should ignore it. A category is any model of the first-order theory of categories. There’s no reason any part of that model needs to be made out of sets. Some authors insist that all categories should have sets of homs between two fixed objects, but they shouldn’t do this. It bothers and confuses people for no good reason.