Common functors as the base of cofree comonads

Cofree Empty ~~ Empty

Given

data Empty a

we have

data Cofree Empty a
   --  = a :< ...  not possible!

Given

data Const c a = Const c

we have

data Cofree (Const c) a
     = a :< Const c

which is isomorphic to

data Writer c a = Writer c a

Given

data Identity a = Identity a

we have

data Cofree Identity a
     = a :< Identity (Cofree Identity a)

which is isomorphic to

data Stream a = Stream a (Stream a)

Given

data Maybe a = Just a
             | Nothing

we have

data Cofree Maybe a
     = a :< Just (Cofree Maybe a)
     | a :< Nothing

which is isomorphic to

data NonEmpty a
     = NECons a (NonEmpty a)
     | NESingle a

Given

data Writer w a = Writer w a

we have

data Cofree (Writer w) a
     = a :< (w, Cofree (Writer w) a)

which is equivalent to

data Stream (w,a)
     = Stream (w,a) (Stream (w,a))

which can properly be written as WriterT w Stream with

data WriterT w m a = WriterT (m (w,a))

Given

data Either e a = Left e
                | Right a

we have

data Cofree (Either e) a
     = a :< Left e
     | a :< Right (Cofree (Either e) a)

which is isomorphic to

data Hospitable e a
     = Sorry_AllIHaveIsThis_Here'sWhy a e
     | EatThis a (Hospitable e a)

or, if you promise to only evaluate the log after the complete result, NonEmptyT (Writer e) a with

data NonEmptyT (Writer e) a = NonEmptyT (e,a,[a])

Given

data Reader x a = Reader (x -> a)

we have

data Cofree (Reader x) a
     = a :< (x -> Cofree (Reader x) a)

which is isomorphic to

data Plant x a
     = Plant a (x -> Plant x a)