There are two ways to define a comonad:

I. Provide definitions for fmap, extract, and duplicate satisfying these laws:

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

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

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

(fmap cannot be defaulted, but a comonad which defines extend may simply set fmap equal to liftW.)

A comonad providing definitions for extend and duplicate, must also satisfy these laws:

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

(The first two are the defaults for extend and duplicate, and the third is the definition of liftW.)