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

module freyd {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) 
  where

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

-- C is small full subcategory of A ( C is image of F )

record SmallFullSubcategory {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) 
      (F : Functor A A ) ( FMap← : { a b : Obj A } → Hom A (FObj F a) (FObj F b ) → Hom A a b )
           : Set  (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where
   field
      ≈→≡ : {a b : Obj A } →  { x y : Hom A (FObj F a) (FObj F b) } → 
                (x≈y : A [ FMap F x ≈ FMap F y  ]) → FMap F x ≡ FMap F y   -- co-comain of FMap is local small
      full→ : { a b : Obj A } { x : Hom A (FObj F a) (FObj F b) } → A [ FMap F ( FMap← x ) ≈ x ]
      full← : { a b : Obj A } { x : Hom A a b } → A [ FMap← ( FMap F x ) ≈ x ]

-- pre-initial

record PreInitial {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) 
      (F : Functor A A )  : Set  (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where
   field
      preinitialObj : ∀{  a : Obj A } →  Obj A 
      preinitialArrow : ∀{  a : Obj A } →  Hom A ( FObj F (preinitialObj {a} )) a 

-- initial object

record HasInitialObject {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (i : Obj A) : Set  (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where
   field
      initial :  ∀( a : Obj A ) →  Hom A i a
      uniqueness  : ( a : Obj A ) →  ( f : Hom A i a ) → A [ f ≈  initial a ]



-- A complete catagory has initial object if it has PreInitial-SmallFullSubcategory

open NTrans
open Limit
open SmallFullSubcategory
open PreInitial

initialFromPreInitialFullSubcategory : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
      (F : Functor A A ) ( FMap← : { a b : Obj A } → Hom A (FObj F a) (FObj F b ) → Hom A a b )
      (lim : ( Γ : Functor A A ) → { a0 : Obj A } { u : NTrans A A ( K A A a0 ) Γ } → Limit A A Γ a0 u ) -- completeness
      (SFS : SmallFullSubcategory A F FMap← ) → 
      (PI : PreInitial A F ) → { a0 : Obj A } → { u : (a : Obj A) → NTrans A A (K A A a) F }
          → HasInitialObject A a0
initialFromPreInitialFullSubcategory A F  FMap← lim SFS PI {a0} {u} = record {
      initial = initialArrow ; 
      uniqueness  = λ a f → lemma1 a f
    } where
      initialArrow :  ∀( a : Obj A )  →  Hom A a0 a
      initialArrow a  = A [ preinitialArrow PI {a}  o limit (lim F {FObj F (preinitialObj PI)} {u (FObj F (preinitialObj PI))} ) a0 (u a0 )  ] 
      lemma1 : (a : Obj A) (f : Hom A a0 a) → A [ f ≈ initialArrow a ]
      lemma1 a f = let open ≈-Reasoning (A) in 
             sym (
             begin
                 initialArrow a
             ≈⟨⟩
                 preinitialArrow PI {a} o  limit (lim F) a0 (u a0) 
             ≈⟨ {!!} ⟩
                 f
             ∎  )