Intuitions on functors.

functors.hs

fmap           :: Functor f           => (a -> b)  -> f a   -> f b

contramap      :: Contravariant f     => (a <- b)  -> f a   -> f b

invmap         :: Invariant f         => (a <-> b) -> f a  <-> f b -- NOT VALID syntax but to give an intuition.

bimap          :: Bifunctor p         => (a -> b)
                                      -> (c -> d)
                                                   -> p a c
                                                   -> p b d

dimap          :: Profunctor p        => (a -> b)
                                      -> (c -> d)
                                                   -> p b c
                                                   -> p a d
                                                   
-- putting bimap and dimap together in a slightly different way:
bimap :: Bifunctor  p => (a -> b) -> (c -> d) -> p a c -> p b d
dimap :: Profunctor p => (a <- b) -> (c -> d) -> p b c -> p a d

-- another way showing the shapes of the types
bimap :: Bifunctor  (p :: * -> *) => (a -> b) -> (c -> d) -> p a c -> p b d
dimap :: Profunctor (p :: * -> *) => (a <- b) -> (c -> d) -> p b c -> p a d

-- Note Bifunctor is covariant in both type arguments
-- Note Profunctor is contravariant in the first type argument and covariant in the second
-- Profunctors are often used for defining decoding and encoding pairs (e.g. JSON (de-)serialization, reading and writing to database).