How many arrow laws am I breaking?

The Power of Arrows

{-# Language TypeOperators #-}

import Prelude hiding ((.), id)
import Control.Arrow
import Control.Category

{-
   > let math = ("add 2" :-: (+2)) . ("times 3" :-: (*3)) >>> ("invert" :-: (1/))
   > math
      (invert) . ((add 2) . (times 3))
   > runFun math 9
      3.4482758620689655e-2
-}

data (:->:) a b = String :-: (a -> b)

runFun :: (a :->: b) -> a -> b
runFun (_ :-: f) a = f a

instance Category (:->:) where
  id = "id" :-: id
  (s1 :-: f1) . (s2 :-: f2) = ("(" ++ s1 ++ ") . (" ++ s2 ++ ")") :-: (f1 . f2)

instance Show (a:->:b) where
  show (s :-: _) = s

instance Arrow (:->:) where
  arr f = "Lambda" :-: f
  first lf@(s :-: f) = ("first (" ++ show lf ++ ")") :-: (\(b, d) -> (f b, d))

I guess I would have to override (&&&) and (***) for the actual arrow behaviour to be meaningful...

This is mainly a Control.Category exercise :D