There are two ways to define a comonad:

I. Provide definitions for extract and extend satisfying these laws:

extend extract      = id
extract . extend f  = f
extend f . extend g = extend (f . extend g)

In this case, you may simply set fmap = liftW.

These laws are directly analogous to the laws for monads and perhaps can be made clearer by viewing them as laws stating that Cokleisli composition must be associative, and has extract for a unit:

f =>= extract   = f
extract =>= f   = f
(f =>= g) =>= h = f =>= (g =>= h)

II. Alternately, you may choose to provide definitions for fmap, extract, and duplicate satisfying these laws:

extract . duplicate      = id
fmap extract . duplicate = id
duplicate . duplicate    = fmap duplicate . duplicate

In this case you may not rely on the ability to define fmap in terms of liftW.

You may of course, choose to define both duplicate and extend. In that case you must also satisfy these laws:

extend f  = fmap f . duplicate
duplicate = extend id
fmap f    = extend (f . extract)

These are the default definitions of extend and duplicate and the definition of liftW respectively.

A suitable default definition for fmap for a Comonad. Promotes a function to a comonad.

fmap f = liftW f = extend (f . extract)

Comonadic fixed point à la Menendez

Comonadic fixed point à la Orchard

Left-to-right Cokleisli composition

Right-to-left Cokleisli composition

extend in operator form

extend with the arguments swapped. Dual to >>= for a Monad.

ComonadApply is to Comonad like Applicative is to Monad.

Mathematically, it is a strong lax symmetric semi-monoidal comonad on the category Hask of Haskell types. That it to say that w is a strong lax symmetric semi-monoidal functor on Hask, where both extract and duplicate are symmetric monoidal natural transformations.

Laws:

(.) <$> u <@> v <@> w = u <@> (v <@> w)
extract (p <@> q) = extract p (extract q)
duplicate (p <@> q) = (<@>) <$> duplicate p <@> duplicate q

If our type is both a ComonadApply and Applicative we further require

(<*>) = (<@>)

Finally, if you choose to define (<@) and (@>), the results of your definitions should match the following laws:

a @> b = const id <$> a <@> b
a <@ b = const <$> a <@> b

A variant of <@> with the arguments reversed.

Lift a binary function into a comonad with zipping

Lift a ternary function into a comonad with zipping

The Cokleisli Arrows of a given Comonad

The Functor class is used for types that can be mapped over. Instances of Functor should satisfy the following laws:

fmap id  ==  id
fmap (f . g)  ==  fmap f . fmap g

The instances of Functor for lists, Maybe and IO satisfy these laws.

An infix synonym for fmap.

Replace the contents of a functor uniformly with a constant value.