-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : Functor.{v, v', u, u'} C D
inst_2 : PreservesColimitsOfShape (Discrete WalkingPair) F
inst_3 : PreservesColimitsOfShape (Discrete PEmpty.{1}) F
inst_4 : HasFiniteCoproducts.{v, u} C
n : ℕ
this_1 : ∀ (f : Fin n → C), PreservesColimit (Discrete.functor f) F
f : Fin (n + 1) → C
this : IsColimit.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
  Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
  (extendCofan
    (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
      F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
    (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
      (F.map coprod.inl) (F.map coprod.inr))) :=
  extendCofanIsColimit (fun (i : Fin (n + 1)) ↦ F.obj (f i))
    (isColimitOfHasCoproductOfPreservesColimit F fun (i : Fin n) ↦ f i.succ)
    (isColimitOfHasBinaryCoproductOfPreservesColimit F (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ))
apply :
  Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F := Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
      (extendCofan
        (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
          F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
        (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
          (F.map coprod.inl) (F.map coprod.inr))) ≅
    Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F := Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
      (Cofan.mk
        (F.obj
          (Cocone.pt.{0, v, 0, u} (F :=
            pair (f 0)
              (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
            (colimit.cocone
              (pair (f 0)
                (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                  (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))))
        fun (j : Fin (n + 1)) ↦
        F.map
          (Fin.cases (motive := fun (x : Fin (n + 1)) ↦
            f x ⟶
              Cocone.pt.{0, v, 0, u} (F :=
                pair (f 0)
                  (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                    (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                (colimit.cocone
                  (pair (f 0)
                    (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                      (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
            (BinaryCofan.inl
              (colimit.cocone
                (pair (f 0)
                  (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                    (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
            (fun (i : Fin n) ↦
              NatTrans.app (Cocone.ι (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))) { as := i } ≫
                BinaryCofan.inr
                  (colimit.cocone
                    (pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
            j))
⊢ Iso (C :=
    Cocone.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (C := D) (Discrete.functor fun (j : Fin (n + 1)) ↦ F.obj (f j)))
    (extendCofan
      (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
        F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
      (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
        (F.map coprod.inl) (F.map coprod.inr)))
    (Cofan.mk
      (F.obj
        (Cocone.pt.{0, v, 0, u} (F :=
          pair (f 0)
            (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
              (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
          (colimit.cocone
            (pair (f 0)
              (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))))
      fun (j : Fin (n + 1)) ↦
      F.map
        (Fin.cases (motive := fun (x : Fin (n + 1)) ↦
          f x ⟶
            Cocone.pt.{0, v, 0, u} (F :=
              pair (f 0)
                (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                  (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
              (colimit.cocone
                (pair (f 0)
                  (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                    (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
          (BinaryCofan.inl
            (colimit.cocone
              (pair (f 0)
                (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                  (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
          (fun (i : Fin n) ↦
            NatTrans.app (Cocone.ι (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))) { as := i } ≫
              BinaryCofan.inr
                (colimit.cocone
                  (pair (f 0)
                    (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                      (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
          j))
have apply₁ : ∀ (j : Discrete (Fin (n + 1))),
  Eq.{v' + 1} (α :=
    Functor.obj.{0, v', 0, u'} (Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i)) j ⟶
      Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F := Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
        (Cofan.mk
          (F.obj
            (Cocone.pt.{0, v, 0, u} (F :=
              pair (f 0)
                (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                  (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
              (colimit.cocone
                (pair (f 0)
                  (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                    (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))))
          fun (j : Fin (n + 1)) ↦
          F.map
            (Fin.cases (motive := fun (x : Fin (n + 1)) ↦
              f x ⟶
                Cocone.pt.{0, v, 0, u} (F :=
                  pair (f 0)
                    (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                      (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                  (colimit.cocone
                    (pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
              (BinaryCofan.inl
                (colimit.cocone
                  (pair (f 0)
                    (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                      (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
              (fun (i : Fin n) ↦
                NatTrans.app (Cocone.ι (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
                    { as := i : Discrete (Fin n) } ≫
                  BinaryCofan.inr
                    (colimit.cocone
                      (pair (f 0)
                        (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                          Discrete.functor fun (i : Fin n) ↦ f i.succ)
                          (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
              j)))
    (CategoryStruct.comp (Y :=
        Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
          Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
          (extendCofan
            (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
              F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
            (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
              (F.map coprod.inl) (F.map coprod.inr))))
        (NatTrans.app
            (Cocone.ι
              (extendCofan
                (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
                  F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
                (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
                  (F.map coprod.inl) (F.map coprod.inr))))
            j :
          Functor.obj.{0, v', 0, u'} (Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i)) j ⟶
            Functor.obj.{0, v', 0, u'}
              (Functor.obj (Functor.const (Discrete (Fin (n + 1))))
                (Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
                  Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
                  (extendCofan
                    (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
                      F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
                    (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
                      (F.map coprod.inl) (F.map coprod.inr)))))
              j)
        (Iso.hom
          (Iso.refl
            (Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
              Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
              (extendCofan
                (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
                  F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
                (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
                  (F.map coprod.inl) (F.map coprod.inr)))))) :
      Functor.obj.{0, v', 0, u'} (Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i)) j ⟶
        Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
          Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
          (Cofan.mk
            (F.obj
              (Cocone.pt.{0, v, 0, u} (F :=
                pair (f 0)
                  (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                    (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                (colimit.cocone
                  (pair (f 0)
                    (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                      (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))))
            fun (j : Fin (n + 1)) ↦
            F.map
              (Fin.cases (motive := fun (x : Fin (n + 1)) ↦
                f x ⟶
                  Cocone.pt.{0, v, 0, u} (F :=
                    pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                    (colimit.cocone
                      (pair (f 0)
                        (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                          Discrete.functor fun (i : Fin n) ↦ f i.succ)
                          (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                (BinaryCofan.inl
                  (colimit.cocone
                    (pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                (fun (i : Fin n) ↦
                  NatTrans.app (Cocone.ι (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
                      { as := i : Discrete (Fin n) } ≫
                    BinaryCofan.inr
                      (colimit.cocone
                        (pair (f 0)
                          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                            Discrete.functor fun (i : Fin n) ↦ f i.succ)
                            (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                j)))
    (NatTrans.app
        (Cocone.ι
          (Cofan.mk
            (F.obj
              (Cocone.pt.{0, v, 0, u} (F :=
                pair (f 0)
                  (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                    (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                (colimit.cocone
                  (pair (f 0)
                    (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                      (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))))
            fun (j : Fin (n + 1)) ↦
            F.map
              (Fin.cases (motive := fun (x : Fin (n + 1)) ↦
                f x ⟶
                  Cocone.pt.{0, v, 0, u} (F :=
                    pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                    (colimit.cocone
                      (pair (f 0)
                        (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                          Discrete.functor fun (i : Fin n) ↦ f i.succ)
                          (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                (BinaryCofan.inl
                  (colimit.cocone
                    (pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                (fun (i : Fin n) ↦
                  NatTrans.app (Cocone.ι (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))) { as := i } ≫
                    BinaryCofan.inr
                      (colimit.cocone
                        (pair (f 0)
                          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                            Discrete.functor fun (i : Fin n) ↦ f i.succ)
                            (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                j)))
        j :
      Functor.obj.{0, v', 0, u'} (Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i)) j ⟶
        Functor.obj.{0, v', 0, u'}
          (Functor.obj (Functor.const (Discrete (Fin (n + 1))))
            (Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
              Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
              (Cofan.mk
                (F.obj
                  (Cocone.pt.{0, v, 0, u} (F :=
                    pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                    (colimit.cocone
                      (pair (f 0)
                        (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                          Discrete.functor fun (i : Fin n) ↦ f i.succ)
                          (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))))
                fun (j : Fin (n + 1)) ↦
                F.map
                  (Fin.cases (motive := fun (x : Fin (n + 1)) ↦
                    f x ⟶
                      Cocone.pt.{0, v, 0, u} (F :=
                        pair (f 0)
                          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                            Discrete.functor fun (i : Fin n) ↦ f i.succ)
                            (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                        (colimit.cocone
                          (pair (f 0)
                            (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                              Discrete.functor fun (i : Fin n) ↦ f i.succ)
                              (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                    (BinaryCofan.inl
                      (colimit.cocone
                        (pair (f 0)
                          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                            Discrete.functor fun (i : Fin n) ↦ f i.succ)
                            (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                    (fun (i : Fin n) ↦
                      NatTrans.app (Cocone.ι (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
                          { as := i : Discrete (Fin n) } ≫
                        BinaryCofan.inr
                          (colimit.cocone
                            (pair (f 0)
                              (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                                Discrete.functor fun (i : Fin n) ↦ f i.succ)
                                (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                    j))))
          j)
Tactic: apply Cocone.ext _ _