Why there is no `Cofunctor` typeclass in Haskell?

Question

Monads get fmap from Functor typeclass. Why comonads don't need a cofmap method defined in a Cofunctor class?

Answer 1

Functor is defined as:

class Functor f where
    fmap :: (a -> b) -> (f a -> f b)

Cofunctor could be defined as follows:

class Cofunctor f where
    cofmap :: (b -> a) -> (f b -> f a)

So, both are technically the same, and that's why Cofunctor does not exist. "The dual concept of 'functor in general' is still 'functor in general'".

Since Functor and Cofunctor are the same, both monads and comonads are defined by using Functor. But don't let that make you think that monads and comonads are the same thing, they're not.

A monad is defined (simplifying) as:

class Functor m => Monad where
    return :: a -> m a
    (>>=) :: m a -> (a -> m b) -> m b

whether a comonad (again, simplified) is:

class Functor w => Comonad where
    extract :: w a -> a
    extend :: (w a -> b) -> w a -> w b

Note the "symmetry".

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Another thing is a contravariant functor, defined as:

import Data.Functor.Contravariant
class Contravariant f where
    contramap :: (b -> a) -> (f a -> f b)