/sci/ - Science & Math

Why does the text require that C be a small category here?
It seems they need it to justify that the natural transformations between any two endofunctors on C form a set; and so the hom-sets of the category End(C) are actual sets.
I believe they do this because their initial definition of a category states that the hom-sets are actual sets. In other texts this isn't required an the hom-sets may very well be proper classes.
So, if we allow the hom-sets of categories to be proper class sized, does this construction still require the smallness condition or may we drop it?
I'm not well versed enough in all this foundational and size issue related stuff