If a functor is representable and its key is a fixed point, then there's a much faster way to implement fmap, pure and ap.

RepresentableFix.hs

{-# LANGUAGE TypeFamilies, DeriveFunctor #-}

import Data.Key (Key)
import Data.Functor.Foldable (Fix(..), fold, unfold)

import Control.Applicative
import Control.Arrow (left)


data RF f a r = RF { getRF :: Key f -> Either a (Key f, r) } deriving Functor
type R f a = Fix (RF f a)

class IndexableFix f where
  indexFix :: f a -> R f a
  
class IndexableFix f => RepresentableFix f where
  tabulateFix :: R f a -> f a

index' :: IndexableFix f => f a -> Key f -> a
index' = fold ((either id (uncurry (flip ($))) .) . getRF) . indexFix

fmapRepFix :: RepresentableFix f => (a -> b) -> f a -> f b
fmapRepFix f = tabulateFix . fold (Fix . RF . fmap (left f) . getRF) . indexFix

pureRepFix :: RepresentableFix f => a -> f a
pureRepFix = tabulateFix . Fix . RF . const . Left

apRepFix :: RepresentableFix f => f (a -> b) -> f a -> f b
apRepFix ff fa = tabulateFix (h (indexFix ff) (indexFix fa))
  where
    h (Fix rf) (Fix ra) = Fix . RF $ \k -> 
      case (getRF rf k, getRF ra k) of
        (Left f, Left x) -> Left (f x) 
        (Right (k', rf'), Right (_, ra')) -> Right (k', h rf' ra')

bindRepFix :: RepresentableFix f => f a -> (a -> f b) -> f b
-- bindRepFix m f = tabulateFix (Fix . RF $ \k -> let Fix (RF rf) = indexFix (f (index' m k)) in rf k) -- does not produce
bindRepFix m f = undefined

data Stream a = a :<: Stream a deriving Show
data Peano = Z | S Peano

type instance Key Stream = Peano

instance IndexableFix Stream where
  indexFix = unfold (RF . f)
    where
      f (a :<: _ )  Z    = Left      a
      f (_ :<: as) (S n) = Right (n, as)

instance RepresentableFix Stream where
  tabulateFix = fold g
    where
      g (RF f) = a :<: as
        where
          Left      a   = f  Z
          Right (n, as) = f (S n)

instance Functor Stream where
  fmap = fmapRepFix

instance Applicative Stream where
  pure = pureRepFix
  (<*>) = apRepFix

tabulate' :: (Peano -> a) -> Stream a
tabulate' = tabulateFix . Fix . RF . g
  where
    g f  Z    = Left  (f Z)
    g f (S n) = Right (n, Fix . RF . g $ f . S)

instance Monad Stream where
  return = pureRepFix
  m >>= f = bindRepFix m f

test :: Stream Int
test = 1 :<: fmap (+1) test

fromPeano :: Peano -> Int
fromPeano Z = 0
fromPeano (S n) = 1 + fromPeano n