{-# OPTIONS --without-K --safe #-}
module Categories.Functor.Hom where
open import Data.Product
open import Function using () renaming (_∘_ to _∙_)
open import Categories.Category
open import Categories.Functor hiding (id)
open import Categories.Functor.Properties
open import Categories.Functor.Bifunctor
open import Categories.Category.Instance.Setoids
import Categories.Morphism.Reasoning as MR
open import Relation.Binary using (Setoid)
module Hom {o ℓ e} (C : Category o ℓ e) where
open Category C
open MR C
Hom[-,-] : Bifunctor (Category.op C) C (Setoids ℓ e)
Hom[-,-] = record
{ F₀ = F₀′
; F₁ = λ where
(f , g) → record
{ to = λ h → g ∘ h ∘ f
; cong = ∘-resp-≈ʳ ∙ ∘-resp-≈ˡ
}
; identity = identityˡ ○ identityʳ
; homomorphism = ∘-resp-≈ʳ sym-assoc ○ assoc²γδ
; F-resp-≈ = λ { (f₁≈g₁ , f₂≈g₂) → f₂≈g₂ ⟩∘⟨ refl⟩∘⟨ f₁≈g₁}
}
where F₀′ : Obj × Obj → Setoid ℓ e
F₀′ (A , B) = hom-setoid {A} {B}
open HomReasoning
Hom[_,-] : Obj → Functor C (Setoids ℓ e)
Hom[_,-] = appˡ Hom[-,-]
Hom[-,_] : Obj → Contravariant C (Setoids ℓ e)
Hom[-,_] = appʳ Hom[-,-]
Hom[_,_] : Obj → Obj → Setoid ℓ e
Hom[ A , B ] = hom-setoid {A} {B}
module _ {o ℓ e} (C : Category o ℓ e) where
open Category C
open Hom C
Hom[_][-,-] : Bifunctor (Category.op C) C (Setoids ℓ e)
Hom[_][-,-] = Hom[-,-]
Hom[_][_,-] : Obj → Functor C (Setoids ℓ e)
Hom[_][_,-] B = Hom[ B ,-]
Hom[_][-,_] : Obj → Contravariant C (Setoids ℓ e)
Hom[_][-,_] B = Hom[-, B ]
Hom[_][_,_] : Obj → Obj → Setoid ℓ e
Hom[_][_,_] A B = hom-setoid {A} {B}