Haskell generalization: is it possible to replace Hask with Cat?

Question

It’s great that Haskell allows us to go into the Hask category. But sometimes I feel that it is too tight. So I had an idea about a programming language that would allow us to move into the Cat category of small categories, where the categories would be types and functors from categories C to D, would be functions like C-> D. At first it seemed like a crazy idea, but I found an article in an article by Glynn Winskel and his student Mario Jose Kakamo about such a calculus! However, it is limited to a Cat fragment. Do you think that it would be possible to expand Haskell in this spirit, that is, instead of working in the Hask category, it will work in the Cat category (or better, 2-category)? Do you know of a calculus similar to that made by Kakamo and Winsel?

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