Lenses, prisms, affine traversals

Lenses and prisms do not compose into affine traversals if one is inclined to believe that Haskell composition of Tambara modules is the definition of composition of optics for optics of different families. I will construct a counterexample that illustrates this fact. I will, however, respect the laws of Tambara modules, and not make use of any freedoms that the limited expressivity of Haskell’s type system would give me. The failure is theoretical, not only a product of this particular implementation.

-- Consider the constantly Bool profunctor.
data CBool a b = CBool Bool
class Profunctor CBool where
  dimap _ _ = id

-- It is a strong profunctor, respecting the laws.
class Strong CBool where
  strong = id