module Cat.Functor.Hom.Duality where

# Duality of Hom functors🔗

We prove the obvious dualities between $\mathbf{Hom}$ functors of opposite categories, and between representable and corepresentable functors.

Hom-from-op : ∀ c → Hom-from (C ^op) c ≡ Hom-into C c Hom-from-op c = Functor-path (λ _ → refl) (λ _ → refl) Hom-into-op : ∀ c → Hom-into (C ^op) c ≡ Hom-from C c Hom-into-op c = Functor-path (λ _ → refl) (λ _ → refl) corepresentable→co-representable : ∀ {F : Functor C (Sets ℓ)} → Corepresentation F → Representation {C = C ^op} F corepresentable→co-representable F-corep .rep = F-corep .corep corepresentable→co-representable F-corep .represents = F-corep .corepresents ∘ni path→iso (sym (Hom-into-op _)) co-representable→corepresentable : ∀ {F : Functor (C ^op) (Sets ℓ)} → Representation {C = C} F → Corepresentation F co-representable→corepresentable F-rep .corep = F-rep .rep co-representable→corepresentable F-rep .corepresents = F-rep .represents ∘ni path→iso (sym (Hom-from-op _))