open import Algebra.G-Set
open import Algebra.Group
open import Category.G-Sets
open import Categories.Category.Instance.Sets
open import Categories.Functor.Core
open import Categories.Category.Core
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
open import Relation.Binary.PropositionalEquality.WithK
open import Axiom.UniquenessOfIdentityProofs.WithK
open import Data.Product
open import Data.Product.Properties renaming (Σ-≡,≡→≡ to peq)
open import Level

module Category.G-Sets.Properties.A4 {ℓ} (G : Group ℓ) where
  open Functor
  open G-Set-Morphism using () renaming (u to <_>)
  D : Functor (Sets ℓ) (G-Sets G)
  D .F₀ S = record
    { X = S
    ; _⊳_ = λ _ s → s
    ; ε⊳ = ≡.refl
    ; ∘⊳ = ≡.refl
    }
  D .F₁ {A} {B} f = record
    { u = f
    ; equivariance = ≡.refl
    }
  D .identity = ≡.refl
  D .homomorphism = ≡.refl
  D .F-resp-≈ f≡g = f≡g

  V : Functor (G-Sets G) (Sets ℓ)
  V .F₀ GS = Orb
    where
      open G-Set GS
  V .F₁ {A} {B} f (x , (y , (g , eq))) = (< f > x , (< f > y , (g , ≡.trans (≡.sym f.equivariance) (≡.cong < f > eq))))
    where
      open G-Set A
      module f = G-Set-Morphism f
  V .identity {GS} {(x , (y , (g , ≡.refl)))} = ≡.refl
  V .homomorphism {X} {Y} {Z} {f} {g} = {!!}
  V .F-resp-≈ {A} {B} {f} {g} eq {(x , (y , (h , ≡.refl)))} = peq (eq , {!!})
    where
      module f = G-Set-Morphism f
      module g = G-Set-Morphism g
      module A = G-Set A