State monad as composition of (a ->) & (, s)

StateMonad.hs

newtype State s a = State { runState :: s -> (a, s) }

instance Functor (State s) where
  --   fmap f st = State
  --     $ \s -> let (a, s') = runState st s
  --             in (f a, s')
  --   fmap f st = State $ fmap (fmap f) (runState st)
  fmap f st = State $ (.) (\(a, s) -> (f a, s)) (runState st)
  fmap f st = State $ (\g x -> (\(a, s) -> (f a, s)) $ g x) (runState st)
  fmap f st = State $ (\x -> (\(a, s) -> (f a, s)) $ (runState st) x)
  fmap f st = State
    $ (\x -> let (a, s) = (runState st) x
             in (f a, s))