FunctorLayer

FunctorLayer[M, Inner] has the following functionality: - lifts values from the Functor Inner to the Functor M. - lifts Functor isomorphisms in Inner ((Inner ~> Inner, Inner ~> Inner)) into Functor homomorphisms in M (M ~> M). This allows you to "map" a natural transformation over the Inner inside M, but only if you can provide an inverse of that natural transformation.

FunctorLayer[M, Inner] has two external laws:

def mapForwardRespectsLayer[A](in: Inner[A])(forward: Inner ~> Inner, backward: Inner ~> Inner) = {
  layer(forward(in)) <-> layerImapK(layer(in))(forward, backward)
}

def layerImapRespectsId[A](in: M[A]) = {
  in <-> layerImapK(in)(FunctionK.id, FunctionK.id)
}

FunctorLayer has one free law, i.e. a law guaranteed by parametricity:

def layerRespectsMap[A, B](in: Inner[A])(f: A => B) = {
  layer(in.map(f)) <-> layer(in).map(f)
} // this is free because all `FunctionK`'s (including `layer`, despite not being an instance of the class)
  // are natural transformations (i.e., they follow this law exactly)