---
--
--  Equalizer
--
--         e             f
--    c  --------> a ----------> b
--    ^        .     ---------->
--    |      .            g
--    |k   .                
--    |  . h              
--    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} (e : Hom  A c a) (h : Hom A d a) →  Hom A d c 
      equalize : {c d : Obj A} (e : Hom  A c a) (h : Hom A d a) →
           A [ A [ A [ f  o  e ] o equalizer e h ]  ≈ A [ g  o h ] ]
      uniqueness : {c d : Obj A} (e : Hom  A c a) (h : Hom A d a) ( k : Hom A d c ) → 
           A [ A [ A [ f  o  e ] o k ]  ≈ A [ g  o h ] ] → A [ equalizer e h  ≈ k ]

record EqEqualizer { 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
      α : {e a b : Obj A}  → (f : Hom A a b) → (g : Hom A a b ) →  Hom A e a
      γ : {c d e a b : Obj A}  → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) →  Hom A c e 
      δ : {e a b : Obj A}  → (f : Hom A a b) → Hom A a e 
      b1 :  {e : Obj A} → A [ A [ f  o α {e} f g ] ≈ A [ g  o α {e} f g ] ]
      b2 :  {c d : Obj A } → {h : Hom A d a } → A [ A [ α {c} f g o γ {c} f g h ] ≈ A [ h  o α  (A [ f o h ]) (A [ g o h ]) ] ]
      b3 :  {e : Obj A} → A [ A [ α {e} f f o δ {e} f ] ≈ id1 A a ]
      -- b4 :  {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o  k ] ) ≈ k ]
      b4 :  {c d : Obj A } {k : Hom A c a} → A [ A [ γ f g ( A [ α f g o  k ] ) o δ {c} (A [ f o A [ α f g o  k ] ] ) ] ≈ k ]
   --  A [ α f g o β f g h ] ≈ h
   β : { d e a b : Obj A}  → (f : Hom A a b) → (g : Hom A a b ) →  (h : Hom A d a ) → Hom A d e
   β {d} f g h =  A [ γ f g h o δ {d} (A [ f o h ]) ] 

lemma-equ1 :  { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ )  {a b : Obj A} (f g : Hom A a b)  → Equalizer A f g → EqEqualizer A f g
lemma-equ1  A {a} {b} f g eqa = record {
      α = {!!} ;
      γ = {!!} ;
      δ = {!!} ;
      b1 = {!!} ;
      b2 = {!!} ;
      b3 = {!!} ;
      b4 = {!!} 
 }