module universal-mapping where

open import Category -- https://github.com/konn/category-agda
open import Level
open import Category.HomReasoning

open Functor
record IsUniversalMapping  {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') 
                 ( U : Functor B A )
                 ( F : Obj A -> Obj B )
                 ( η : (a : Obj A) -> Hom A a ( FObj U (F a) ) )
                 ( _* : { a : Obj A}{ b : Obj B} -> ( Hom A a (FObj U b) ) ->  Hom B (F a ) b )
                 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where
   field
       universalMapping :   {a' : Obj A} { b' : Obj B } -> ( f : Hom A a' (FObj U b') ) -> 
                     A [ A [ FMap U ( f * ) o η a' ]  ≈ f ]


record UniversalMapping  {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') 
                 ( U : Functor B A )
                 ( F : Obj A -> Obj B )
                 ( η : (a : Obj A) -> Hom A a ( FObj U (F a) ) )
                 ( _* : { a : Obj A}{ b : Obj B} -> ( Hom A a (FObj U b) ) ->  Hom B (F a ) b )
                 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where
    field
       isUniversalMapping : IsUniversalMapping A B U F η _*