What would be the methods of a bi-comonad?

Question

While musing what more useful standard class to suggest to this one

class Coordinate c where
  createCoordinate :: x -> y -> c x y
  getFirst :: c x y ->         


        

Answer 1

This is not an answer, but for Bimonad, how about this?

class Biapplicative p => Bimonad p where
  (>>==) :: p a b -> (a -> b -> p c d) -> p c d

biap :: Bimonad p => p (a -> b) (c -> d) -> p a c -> p b d
biap p q = p >>== \ab cd -> q >>== \a c -> bipure (ab a) (cd c)

instance Bimonad (,) where
  (a,b) >>== f = f a b

I don't know if this is categorically right/interesting, or even remotely useful, but it smells right from a Haskell perspective. Does it match your Bicomonad or something similar?