Lift a function for multicomposition. This is like the :: of a type signature.

Multicompose a function on the appropriate argument. This is like the -> arrows in a type signature.

Multicompose a function on the appropriate argument, calling the left function eagerly. That is, the resulting function will be strict in a if the left argument is strict in a (assuming the final function of the multicomposition, the one applied to the return value, is also strict).

Binary composition: pass two args to the right argument before composing.

 (f .: g) x y = f (g x y)

or,

 f .: g = curry (f . uncurry g)

This is the same as the common idiom (f .) . g but more easily extended to multiple uses, due to the fixity declaration.

Secondary composition: compose the right argument on the second arg of the left argument.

 (f .^ g) x y = f x (g y)

Function composition which calls the right-hand function eagerly; i.e., making the left-hand function strict in its first argument.

 (f .! g) x = f $! g x

This defines the composition for the sub-category of strict Haskell functions. If the Functor class were parameterized by the domain and codomain categories (e.g., a regular Functor f would be CFunctor (->) (->) f instead) then this would allow us to define functors CFunctor (->) (!->) f where fmap f . fmap g = fmap (f .! g).