# # Common functors as the base of cofree comonads

## # Cofree Empty ~~ Empty

Given

```
data Empty a
```

we have

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

## # Cofree (Const c) ~~ Writer c

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
```

## # Cofree Identity ~~ Stream

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)
```

## # Cofree Maybe ~~ NonEmpty

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
```

## # Cofree (Writer w) ~~ WriterT w Stream

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))
```

## # Cofree (Either e) ~~ NonEmptyT (Writer e)

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])
```

## # Cofree (Reader x) ~~ Moore x

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)
```

aka Moore machine.