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.

Yandex:

First result with a more up to date version. No AI, only quotes from the source. Still shows a bunch of SEO.