--- a/limit-to.agda	Sun Mar 06 00:23:35 2016 +0900
+++ b/limit-to.agda	Sun Mar 06 01:05:03 2016 +0900
@@ -29,18 +29,25 @@
    t0 : TwoObject 
    t1 : TwoObject 
 
-record TwoCat  {ℓ c₁ c₂ : Level  } (A : Category  c₁ c₂ ℓ) : Set  (c₂ ⊔ c₁ ⊔ ℓ)  where
+record TwoCat  {ℓ c₁ c₂ : Level  } (A : Category  c₁ c₂ ℓ)  ( a b : Obj A ) ( f g : Hom A a b ): Set  (c₂ ⊔ c₁ ⊔ ℓ)  where
     field
          obj→ : Obj A  -> TwoObject 
          hom→ : {a b : Obj A} -> Hom A a b -> TwoObject
+    fobj : Obj A -> Obj A
+    fobj x with  obj→ x
+    ... | t0 = a
+    ... | t1 = b
+    fmap' :   TwoObject -> Hom A a b
+    fmap' t0 = f
+    fmap' t1 = g
 
 open TwoCat
 
-indexFunctor :  {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) -> (two : TwoCat A) ->
-        ( a b : Obj A ) ( f g : Hom A a b ) ( f-rev : Hom A b a ) 
-        -> Functor A A
-indexFunctor A two a b f g f-rev = record {
-         FObj = λ a → fobj a
+indexFunctor :  {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) 
+        ( a b : Obj A ) ( f g : Hom A a b ) ( f-rev : Hom A b a ) -> (two : TwoCat A a b f g ) 
+            -> Functor A A
+indexFunctor A a b f g f-rev two = record {
+         FObj = λ a → fobj two a
        ; FMap = λ f → fmap f
        ; isFunctor = record {
              identity = {!!} 
@@ -48,37 +55,30 @@
              ; ≈-cong = {!!}
        }
       } where
-          fobj : Obj A -> Obj A
-          fobj x with  obj→ two x
-          ... | t0 = a
-          ... | t1 = b
-          fmap' :  (x y : Obj A) -> TwoObject -> Hom A a b
-          fmap' _ _ t0 = f
-          fmap' _ _ t1 = g
-          fmap :  {x y : Obj A } -> Hom A x y -> Hom A (fobj x) (fobj y)
+          fmap :  {x y : Obj A } -> Hom A x y -> Hom A (fobj two x) (fobj two y)
           fmap {x} {y} f' with obj→ two x | obj→ two y 
           ... | t0 | t0 = id1 A a
           ... | t1 | t0 = f-rev
           ... | t1 | t1 = id1 A b
-          ... | t0 | t1  = fmap' x y (hom→ two f')
+          ... | t0 | t1  = fmap' two (hom→ two f')
 
 open Limit
 
-lim-to-equ : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (two : TwoCat A ) 
+lim-to-equ : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) 
       (lim : (I : Category c₁ c₂ ℓ) ( 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-rev : Hom A b a ) (f g : Hom A  a b )
+        →  {a b c : Obj A}  (f-rev : Hom A b a ) (f g : Hom A  a b ) (two : TwoCat A a b f g ) 
         → (e : Hom A c a ) → (fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] ) → Equalizer A e f g
-lim-to-equ A two lim {a} {b} {c} f-rev f g e fe=ge = record {
+lim-to-equ A lim {a} {b} {c} f-rev f g two 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
-         Γ = indexFunctor A two a b f g f-rev
+         Γ = indexFunctor A a b f g f-rev two
          nmap :  (x : Obj A) ( d : Obj A ) (h : Hom A d a ) -> Hom A (FObj (K A A d) x) (FObj Γ x)
          nmap x d h with (obj→  two x)
          ... | t0 = h
-         ... | t1 = A [ f  o  h ]
+         ... | t1 = A [ fmap' two (obj→  two x) o  h ]
          commute1 : {x y : Obj A}  {f' : Hom A x y} (d : Obj A) (h : Hom A d a ) ->  A [ A [ f  o  h ] ≈ A [ g  o h ] ] -> TwoObject -> TwoObject 
                  → A [ A [ FMap Γ f' o nmap x d h ] ≈ A [ nmap y d h o FMap (K A A d) f' ] ]
          commute1  {x} {y} {f'} d h fh=gh t0 t0 =   let open ≈-Reasoning (A) in