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

module limit-to {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) 
  where

open import cat-utility
open import HomReasoning
open import Relation.Binary.Core
open import Category.Sets
open Functor

open import Data.Nat hiding ( _⊔_ ) renaming ( suc to succ )
open import Data.Bool

open import Data.Fin as Fin
  using (Fin; Fin′; #_; toℕ) renaming (_ℕ-ℕ_ to _-_ ; zero to Fzero ; suc to Fsuc )



-- If we have limit then we have equalizer                                                                                                                                                                  
---  two objects category
---
---          f
---       ------>
---     0         1
---       ------>
---          g


data TwoObject {c₁}  : Set c₁ where
   t0 : TwoObject 
   t1 : TwoObject 


record TwoCat  {ℓ c₁ c₂ : Level  } (Two : Category  c₁ c₂ ℓ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ)) where
    field
         hom→ : Obj Two  -> TwoObject {ℓ}
         hom← : TwoObject {ℓ} -> Obj Two
         hom-iso : {a : Obj Two} -> hom← ( hom→ a) ≡ a
         hom-rev : {a : TwoObject {ℓ} } -> hom→ ( hom← a) ≡ a


open Limit

lim-to-equ : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (I : Category c₁ c₂ ℓ)
      (two : TwoCat A ) 
      (lim : ( Γ : Functor I A ) → { a0 : Obj A } { u : NTrans I A ( K A I a0 ) Γ } → Limit A I Γ a0 u ) -- completeness
        →  {a b c : Obj A} (f g : Hom A a b)  → (e : Hom A c a ) → (fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] ) → Equalizer A e f g
lim-to-equ {c₁} A I two lim {a} {b} {c} f g e fe=ge = record {
        fe=ge =  fe=ge
        ; k = λ {d} h fh=gh → k {d} h fh=gh
        ; ek=h = λ {d} {h} {fh=gh} → {!!}
        ; uniqueness = λ {d} {h} {fh=gh} {k'} → {!!}
     } where
         Γobj :  {!!}  → Obj A
         Γobj t0 = a
         Γobj t1 = b
         Γmap :  {!!}  → Hom A a b
         Γmap t0 = f
         Γmap t1 = g
         Γ : Functor I A
         Γ = record {
            FObj  = λ x → Γobj {!!} ;
            FMap  = λ f → {!!} ;
            isFunctor = record {
                     ≈-cong  = {!!} ; 
                    identity = {!!} ;
                    distr = {!!}
            }
          }
         nat : (d : Obj A) → NTrans I A (K A I d) Γ
         nat d = record {
            TMap = λ x → {!!} ;
            isNTrans = record {
                commute = {!!}
            }
          }
         k : {d : Obj A}  (h : Hom A d a) → A [ A [ f  o  h ] ≈ A [ g  o h ] ] → Hom A d c
         k {d} h fh=gh  = limit (lim Γ {c} {nat c}) d (nat d)