Monoid is a one object Category

M.hs

{-# LANGUAGE NoImplicitPrelude, GADTs, DataKinds, PolyKinds, FlexibleInstances #-}

import Data.Monoid
import Control.Category

-- A monoid gives a rise to a single-object category.
newtype M m (a :: ()) (b :: ()) = M m

instance Monoid m => Category (M m) where
  id = M mempty
  M a . M b = M (mappend a b)

-- A category of a single object is a monoid.
instance (Category c, a ~ '(), b ~ '()) => Monoid (c a b) where
  mempty = id
  mappend = (.)

-- Since we can go both ways and lose nothing along the way, we conclude
-- that a Monoid *is*, in fact, a one object category.