Data.Biapplicative

A bifunctor is a type constructor that takes two type arguments and is a functor in both arguments. That is, unlike with Functor, a type constructor such as Either does not need to be partially applied for a Bifunctor instance, and the methods in this class permit mapping functions over the Left value or the Right value, or both at the same time.

Formally, the class Bifunctor represents a bifunctor from Hask -> Hask.

Intuitively it is a bifunctor where both the first and second arguments are covariant.

The class definition of a Bifunctor p uses the QuantifiedConstraints language extension to quantify over the first type argument a in its context. The context requires that p a must be a Functor for all a. In other words a partially applied Bifunctor must be a Functor. This makes Functor a superclass of Bifunctor such that a function with a Bifunctor constraint may use fmap in its implementation. Functor has been a quantified superclass of Bifunctor since base-4.18.0.0.

You can define a Bifunctor by either defining bimap or by defining both first and second. The second method must agree with fmap:

second ≡ fmap

From this it follows that:

second id ≡ id

If you supply bimap, you should ensure that:

bimap id id ≡ id

If you supply first and second, ensure:

first id ≡ id
second id ≡ id

If you supply both, you should also ensure:

bimap f g ≡ first f . second g

These ensure by parametricity:

bimap  (f . g) (h . i) ≡ bimap f h . bimap g i
first  (f . g) ≡ first  f . first  g
second (f . g) ≡ second f . second g