A natural transformation is a concept from category theory for a mapping between two functors and their objects that preserves a naturality condition. In Haskell the naturality condition boils down to parametricity, so a natural transformation between two functors f and g is represented as

type NaturalTransformation f g = ∀a. f a → g a

This type appears in several Haskell libraries, most obviously in natural-transformations. There are times, however, when we crave more control. Sometimes what we want to do depends on which type a is hiding in that f a we're given. Sometimes, in other words, we need an unnatural transformation.

This means we have to abandon parametricity for ad-hoc polymorphism, and that means type classes. There are two steps to defining a transformation:

• an instance of the base class Transformation declares the two functors being mapped, much like a function type signature,
• while the actual mapping of values is performed by an arbitrary number of instances of the method $, a bit like multiple equation clauses that make up a single function definition.

The module is meant to be imported qualified.