Fast and loose reasoning as to why Functor and Cofunctor are the same. I have to answer this when coworkers as why Comonads are also Functors and not Cofunctors.

Functor.purs

-- Informally showing that Cofunctor is essentially the same structure as Functor thus why Comonads are a Functor which is 
-- effectively a Cofunctor.

-- In PureScript Functor is defined here: https://pursuit.purescript.org/packages/purescript-prelude/4.1.1/docs/Data.Functor#t:Functor
-- class Functor (f :: Type -> Type) where
--   map :: forall a b. (a -> b) -> f a -> f b

-- Informally: Making a dual of something is basically flipping the arrows. But to make is easier to see, let's flip the 
-- arguments of map in Functor definition since that shouldn't change anything, like so:

class Functor' (f :: Type -> Type) where
  mapFlipped :: forall a b. f a -> (a -> b) -> f b

-- Since flipping order of arguments doesn't structurally change things then Functor' is essentially equivalent to Functor 
-- in Prelude (see link above).

class Cofunctor' (f :: Type -> Type) where
  comapFlipped :: forall b a. f b -> (b -> a) -> f a
  -- swapping type variables of a for b and b for a gives us:
  -- forall a b. f a -> (a -> b) -> f b
  -- which is mapFlipped

-- Flipping order of arguments doesn't structurally change things therefore Cofunctor' is equivalent to Cofunctor
-- and since Functor' === Cofunctor' then it follows that Functor === Cofunctor.