both: Like Maybe, but with a different Monoid instance.

The Monoid instance for Maybe behaves like so:

instance Monoid a => Monoid (Maybe a) where
  mappend (Just x) (Just y) = Just $ x <> y
  mappend (Just x) Nothing  = Just x
  mappend Nothing  (Just y) = Just y
  mappend Nothing  Nothing  = Nothing

  mempty = Nothing

Both is a newtype wrapper around Maybe providing this instance:

instance Monoid a => Monoid (Both a) where
  mappend (Just x) (Just y) = Just $ x <> y
  mappend _ _ = Nothing

  mempty = Just mempty

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