FunctorLayerFunctor

FunctorLayerFunctor is the capability to lift Functor homomorphisms in Inner (Inner ~> Inner) into homomorphisms in M (Inner ~> Inner).

FunctorLayerFunctor[M, Inner] has two external laws:

def layerMapRespectsLayerImapK[A](ma: M[A])(forward: Inner ~> Inner,
                                         backward: Inner ~> Inner) = {
  layerImapK(ma)(forward, backward) <-> layerMapK(ma)(forward)
}

def layerMapRespectsLayer[A](in: Inner[A])(forward: Inner ~> Inner) = {
  layer(forward(in)) <-> layerMapK(layer(in))(forward)
}

FunctorLayerFunctor[M, Inner] has one free law, that is, one law guaranteed by other laws and parametricity:

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