---
--
--  Equalizer
--
--         f'            f
--    c  --------> a ----------> b
--    |        .     ---------->
--    |      .            g
--    |h   .                
--    v  .  g'            
--    d 
--
--                        Shinji KONO <kono@ie.u-ryukyu.ac.jp>
----

open import Category -- https://github.com/konn/category-agda                                                                                     
open import Level
open import Category.Sets
module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where

open import HomReasoning
open import cat-utility

record Equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ }  {a b : Obj A} (f g : Hom A a b) : Set  ( ℓ ⊔ (c₁ ⊔ c₂)) where
   field
      equalizer : {c d : Obj A} (f' : Hom  A c a) (g' : Hom A d a) →  Hom A c d 
      equalize : {c d : Obj A} (f' : Hom  A c a) (g' : Hom A d a) →
           A [ A [ f  o  f' ]  ≈ A [  A [ g  o g' ] o equalizer f' g' ] ]
      uniqueness : {c d : Obj A} (f' : Hom  A c a) (g' : Hom A d a) ( e : Hom A c d ) → 
           A [ A [ f  o  f' ]  ≈ A [  A [ g  o g' ] o e ] ] → A [ e  ≈ equalizer f' g' ]