Generalizing applicative and Monad for more categories

Question

In Haskell, we have functions:

fmap :: Functor f => (a -> b) -> (f a -> f b)
pure :: Applicative f => a -> f a
(<*>) :: Applicative f => f (a -> b) -> (f a -> f b)
(=<<) :: Monad f => (a -> f b) -> (f a -> f b)

I can write fmapin terms pureand (<*>)as follows:

fmap x = (<*>) (pure x)

I can also write (<*>)in terms (=<<)and pureas follows:

m1 <*> m2 = (\x -> (\y -> pure (x y)) =<< m2) =<< m1

Having noticed earlier, I threw the binding operator from normal (>>=)to (=<<). The reason I did this is because now I want to generalize such functions:

fmap :: (Functor f, Category cat) => cat a b -> cat (f a) (f b)
pure :: (Applicative f, Category cat) => a -> f a
(<*>) :: (Applicative f, Category cat) => f (cat a b) -> cat (f a) (f b)
(=<<) :: (Monad f, Category cat) => cat a (f b) -> cat (f a) (f b)

Interestingly, I can still write fmapin terms pureand in (<*>)exactly the same way as before:

fmap x = (<*>) (pure x)

, (<*>) pure (=<<). ? , , , (<*>) pure (=<<) , . , .

, , , :

class Category cat => SpecialCategory cat where
  ...

instance SpecialCategory (->) where
  ...

instance SpecialCategory SomeOtherCategory where
  ...

, (<*>) (=<<) pure.

? ?