open import Categories.Category.Core
open import Algebra.Group
open import Data.Unit.Polymorphic
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)

module Category.Construction.GroupAsCategory {ℓ} (G : Group ℓ) where
  open Group G

  GroupAsCategory : Category ℓ ℓ ℓ
  GroupAsCategory = record
                     { Obj = ⊤
                     ; _⇒_ = λ _ _ → Carrier
                     ; _≈_ = _≡_
                     ; id = ε
                     ; _∘_ = _∙_
                     ; assoc = assoc
                     ; sym-assoc = ≡.sym assoc
                     ; identityˡ = idˡ
                     ; identityʳ = idʳ
                     ; identity² = idˡ
                     ; equiv = record
                       { refl = ≡.refl
                       ; sym = ≡.sym
                       ; trans = ≡.trans
                       }
                     ; ∘-resp-≈ = ∙-resp
                     }