module MonadK {o ℓ e} (D : ExtensiveDistributiveCategory o ℓ e) where
  open ExtensiveDistributiveCategory D renaming (U to C; id to idC)
  open Equiv

  -- TODO move this to a different file

  forgetfulF : Functor (Uniform-Iteration-Algebras D) C
  forgetfulF = record
    { F₀ = λ X → Uniform-Iteration-Algebra.A X
    ; F₁ = λ f → Uniform-Iteration-Algebra-Morphism.h f
    ; identity = refl
    ; homomorphism = refl
    ; F-resp-≈ = id
    }

  -- typedef
  FreeUniformIterationAlgebra : Obj → Set (suc o ⊔ suc ℓ ⊔ suc e)
  FreeUniformIterationAlgebra X = FreeObject {C = C} {D = Uniform-Iteration-Algebras D} forgetfulF X

  record MonadK : Set (suc o ⊔ suc ℓ ⊔ suc e) where
    field
      algebras : ∀ X → FreeUniformIterationAlgebra X

    freeF : Functor C (Uniform-Iteration-Algebras D)
    freeF = FO⇒Functor forgetfulF algebras
    
    adjoint : freeF ⊣ forgetfulF
    adjoint = FO⇒LAdj forgetfulF algebras

    K : Monad C
    K = adjoint⇒monad adjoint

    -- EilenbergMoore⇒UniformIterationAlgebras : StrongEquivalence (EilenbergMoore K) (Uniform-Iteration-Algebras D)
    -- EilenbergMoore⇒UniformIterationAlgebras = {!   !}