Doesn’t that construction only work in categories that also contain their own morphisms as objects since a profunctor maps (Cᵒᵖ × C) → Set and not the same like (Cᵒᵖ × C) → C? Since the category of Haskell types special, containing its own morphisms, so the profunctor could be like (haskᵒᵖ × hask) -> hask? or I just don’t understand it.
Hom functors exist for locally small categories, which is just to say that the hom classes are sets. The distinction can be ignored often because local smallness is a trivial consequence of how the category is defined, but it’s not generally true
Doesn’t that construction only work in categories that also contain their own morphisms as objects since a profunctor maps
(Cᵒᵖ × C) → Setand not the same like(Cᵒᵖ × C) → C? Since the category of Haskell types special, containing its own morphisms, so the profunctor could be like(haskᵒᵖ × hask) -> hask? or I just don’t understand it.Hom functors exist for locally small categories, which is just to say that the hom classes are sets. The distinction can be ignored often because local smallness is a trivial consequence of how the category is defined, but it’s not generally true