Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
F : C ⥤ D
inst✝² : PreservesLimitsOfShape (Discrete WalkingPair) F
inst✝¹ : PreservesLimitsOfShape (Discrete PEmpty.{1}) F
inst✝ : HasFiniteProducts C
n : ℕ
this✝ : ∀ (f : Fin n → C), PreservesLimit (Discrete.functor f) F
f : Fin (n + 1) → C
this : IsLimit
  (extendFan (Fan.mk (F.obj (∏ᶜ fun i ↦ f i.succ)) fun j ↦ F.map (Pi.π (fun i ↦ f i.succ) j))
    (BinaryFan.mk (F.map prod.fst) (F.map prod.snd))) :=
  extendFanIsLimit (fun i ↦ F.obj (f i)) (isLimitOfHasProductOfPreservesLimit F fun i ↦ f i.succ)
    (isLimitOfHasBinaryProductOfPreservesLimit F (f 0) (∏ᶜ fun i ↦ f i.succ))
⊢ extendFan (Fan.mk (F.obj (∏ᶜ fun i ↦ f i.succ)) fun j ↦ F.map (Pi.π (fun i ↦ f i.succ) j))
      (BinaryFan.mk (F.map prod.fst) (F.map prod.snd)) ≅
    Fan.mk (F.obj (limit.cone (pair (f 0) (limit.cone (Discrete.functor fun i ↦ f i.succ)).pt)).pt) fun j ↦
      F.map
        (Fin.cases (BinaryFan.fst (limit.cone (pair (f 0) (limit.cone (Discrete.functor fun i ↦ f i.succ)).pt)))
          (fun i ↦
            BinaryFan.snd (limit.cone (pair (f 0) (limit.cone (Discrete.functor fun i ↦ f i.succ)).pt)) ≫
              (limit.cone (Discrete.functor fun i ↦ f i.succ)).π.app { as := i })
          j)

Tactic:
apply Cone.ext _ _

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
F : C ⥤ D
inst✝² : PreservesLimitsOfShape (Discrete WalkingPair) F
inst✝¹ : PreservesLimitsOfShape (Discrete PEmpty.{1}) F
inst✝ : HasFiniteProducts C
n : ℕ
this✝ : ∀ (f : Fin n → C), PreservesLimit (Discrete.functor f) F
f : Fin (n + 1) → C
this : IsLimit
  (extendFan (Fan.mk (F.obj (∏ᶜ fun i ↦ f i.succ)) fun j ↦ F.map (Pi.π (fun i ↦ f i.succ) j))
    (BinaryFan.mk (F.map prod.fst) (F.map prod.snd))) := ⋯
⊢ (extendFan (Fan.mk (F.obj (∏ᶜ fun i ↦ f i.succ)) fun j ↦ F.map (Pi.π (fun i ↦ f i.succ) j))
        (BinaryFan.mk (F.map prod.fst) (F.map prod.snd))).pt ≅
    (Fan.mk (F.obj (limit.cone (pair (f 0) (limit.cone (Discrete.functor fun i ↦ f i.succ)).pt)).pt) fun j ↦
        F.map
          (Fin.cases (BinaryFan.fst (limit.cone (pair (f 0) (limit.cone (Discrete.functor fun i ↦ f i.succ)).pt)))
            (fun i ↦
              BinaryFan.snd (limit.cone (pair (f 0) (limit.cone (Discrete.functor fun i ↦ f i.succ)).pt)) ≫
                (limit.cone (Discrete.functor fun i ↦ f i.succ)).π.app { as := i })
            j)).pt

C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
F : C ⥤ D
inst✝² : PreservesLimitsOfShape (Discrete WalkingPair) F
inst✝¹ : PreservesLimitsOfShape (Discrete PEmpty.{1}) F
inst✝ : HasFiniteProducts C
n : ℕ
this✝ : ∀ (f : Fin n → C), PreservesLimit (Discrete.functor f) F
f : Fin (n + 1) → C
this : IsLimit
  (extendFan (Fan.mk (F.obj (∏ᶜ fun i ↦ f i.succ)) fun j ↦ F.map (Pi.π (fun i ↦ f i.succ) j))
    (BinaryFan.mk (F.map prod.fst) (F.map prod.snd))) := ⋯
⊢ ∀ (j : Discrete (Fin (n + 1))),
    (extendFan (Fan.mk (F.obj (∏ᶜ fun i ↦ f i.succ)) fun j ↦ F.map (Pi.π (fun i ↦ f i.succ) j))
              (BinaryFan.mk (F.map prod.fst) (F.map prod.snd))).π.app
        j =
      Iso.hom ?m.163 ≫
        (Fan.mk (F.obj (limit.cone (pair (f 0) (limit.cone (Discrete.functor fun i ↦ f i.succ)).pt)).pt) fun j ↦
                F.map
                  (Fin.cases
                    (BinaryFan.fst (limit.cone (pair (f 0) (limit.cone (Discrete.functor fun i ↦ f i.succ)).pt)))
                    (fun i ↦
                      BinaryFan.snd (limit.cone (pair (f 0) (limit.cone (Discrete.functor fun i ↦ f i.succ)).pt)) ≫
                        (limit.cone (Discrete.functor fun i ↦ f i.succ)).π.app { as := i })
                    j)).π.app
          j