-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.Over.Products (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
X Y Z : C
f : Y ⟶ X
g : Z ⟶ X
X_1 Y_1 : BinaryFan (Over.mk f) (Over.mk g)
f_1 : X_1 ⟶ Y_1
⊢ Eq (α :=
    Functor.obj
        (Functor.comp (D := PullbackCone f g)
          {
            obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
              PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) ⋯,
            map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
              { hom := Over.Hom.left a.hom, w := ⋯ },
            map_id := ⋯, map_comp := ⋯ }
          {
            obj := fun (c : PullbackCone f g) ↦
              BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst ⋯) (Over.homMk c.snd ⋯),
            map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦ { hom := Over.homMk a.hom ⋯, w := ⋯ }, map_id := ⋯,
            map_comp := ⋯ })
        X_1 ⟶
      Functor.obj (𝟭 (BinaryFan (Over.mk f) (Over.mk g))) Y_1)
    (Functor.map
        (Functor.comp (D := PullbackCone f g)
          {
            obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
              PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) ⋯,
            map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
              { hom := Over.Hom.left a.hom, w := ⋯ :
                ConeMorphism (PullbackCone.mk (Over.Hom.left c₁.fst) (Over.Hom.left c₁.snd) ⋯)
                  (PullbackCone.mk (Over.Hom.left c₂.fst) (Over.Hom.left c₂.snd) ⋯) },
            map_id := ⋯, map_comp := ⋯ : BinaryFan (Over.mk f) (Over.mk g) ⥤ PullbackCone f g }
          {
            obj := fun (c : PullbackCone f g) ↦
              BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst ⋯) (Over.homMk c.snd ⋯),
            map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦
              { hom := Over.homMk a.hom ⋯, w := ⋯ :
                ConeMorphism (C := Over.{v, u} X) (F := pair (Over.mk f) (Over.mk g))
                  (BinaryFan.mk (P := Over.mk (c₁.fst ≫ f)) (Over.homMk c₁.fst ⋯) (Over.homMk c₁.snd ⋯))
                  (BinaryFan.mk (P := Over.mk (c₂.fst ≫ f)) (Over.homMk c₂.fst ⋯) (Over.homMk c₂.snd ⋯)) },
            map_id := ⋯, map_comp := ⋯ : PullbackCone f g ⥤ BinaryFan (Over.mk f) (Over.mk g) })
        f_1 ≫
      Iso.hom
        ((fun (X_2 : BinaryFan (Over.mk f) (Over.mk g)) ↦
            (BinaryFan.ext
                (Over.isoMk
                  (Iso.refl
                    (Over.left.{v, u} (X := X)
                      (Cone.pt (J := Discrete WalkingPair) (F := pair (Over.mk f) (Over.mk g))
                        (Functor.obj
                          (Functor.comp (D := PullbackCone f g)
                            {
                              obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
                                PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) ⋯,
                              map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
                                { hom := Over.Hom.left a.hom, w := ⋯ },
                              map_id := ⋯, map_comp := ⋯ }
                            {
                              obj := fun (c : PullbackCone f g) ↦
                                BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst ⋯) (Over.homMk c.snd ⋯),
                              map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦
                                { hom := Over.homMk a.hom ⋯, w := ⋯ },
                              map_id := ⋯, map_comp := ⋯ })
                          X_2))))
                  ⋯)
                ⋯ ⋯ :
              Functor.obj
                  (Functor.comp (D := PullbackCone f g)
                    {
                      obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
                        PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) ⋯,
                      map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
                        { hom := Over.Hom.left a.hom, w := ⋯ },
                      map_id := ⋯, map_comp := ⋯ }
                    {
                      obj := fun (c : PullbackCone f g) ↦
                        BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst ⋯) (Over.homMk c.snd ⋯),
                      map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦ { hom := Over.homMk a.hom ⋯, w := ⋯ },
                      map_id := ⋯, map_comp := ⋯ })
                  X_2 ≅
                Functor.obj (𝟭 (BinaryFan (Over.mk f) (Over.mk g))) X_2))
          Y_1))
    (Iso.hom
        ((fun (X_2 : BinaryFan (Over.mk f) (Over.mk g)) ↦
            (BinaryFan.ext
                (Over.isoMk
                  (Iso.refl
                    (Over.left.{v, u} (X := X)
                      (Cone.pt (J := Discrete WalkingPair) (F := pair (Over.mk f) (Over.mk g))
                        (Functor.obj
                          (Functor.comp (D := PullbackCone f g)
                            {
                              obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
                                PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) ⋯,
                              map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
                                { hom := Over.Hom.left a.hom, w := ⋯ },
                              map_id := ⋯, map_comp := ⋯ }
                            {
                              obj := fun (c : PullbackCone f g) ↦
                                BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst ⋯) (Over.homMk c.snd ⋯),
                              map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦
                                { hom := Over.homMk a.hom ⋯, w := ⋯ },
                              map_id := ⋯, map_comp := ⋯ })
                          X_2))))
                  ⋯)
                ⋯ ⋯ :
              Functor.obj
                  (Functor.comp (D := PullbackCone f g)
                    {
                      obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
                        PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) ⋯,
                      map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
                        { hom := Over.Hom.left a.hom, w := ⋯ },
                      map_id := ⋯, map_comp := ⋯ }
                    {
                      obj := fun (c : PullbackCone f g) ↦
                        BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst ⋯) (Over.homMk c.snd ⋯),
                      map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦ { hom := Over.homMk a.hom ⋯, w := ⋯ },
                      map_id := ⋯, map_comp := ⋯ })
                  X_2 ≅
                Functor.obj (𝟭 (BinaryFan (Over.mk f) (Over.mk g))) X_2))
          X_1) ≫
      Functor.map (𝟭 (BinaryFan (Over.mk f) (Over.mk g))) f_1)
have w : Eq (α :=
  Over.left.{v, u} (X := X)
      (Cone.pt (J := Discrete WalkingPair) (F := pair (Over.mk f) (Over.mk g))
        (Functor.obj
          (Functor.comp (D := PullbackCone f g)
            {
              obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
                PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) sorry,
              map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
                { hom := Over.Hom.left a.hom, w := sorry },
              map_id := sorry, map_comp := sorry }
            {
              obj := fun (c : PullbackCone f g) ↦
                BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst sorry) (Over.homMk c.snd sorry),
              map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦ { hom := Over.homMk a.hom sorry, w := sorry }, map_id := sorry,
              map_comp := sorry })
          X_1)) ⟶
    (Functor.obj (𝟭 (BinaryFan (Over.mk f) (Over.mk g))) Y_1).pt.left)
  (Over.Hom.left
    (ConeMorphism.hom
      (Functor.map
          (Functor.comp (D := PullbackCone f g)
            {
              obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
                PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) sorry,
              map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
                { hom := Over.Hom.left a.hom, w := sorry :
                  ConeMorphism (PullbackCone.mk (Over.Hom.left c₁.fst) (Over.Hom.left c₁.snd) sorry)
                    (PullbackCone.mk (Over.Hom.left c₂.fst) (Over.Hom.left c₂.snd) sorry) },
              map_id := sorry, map_comp := sorry : BinaryFan (Over.mk f) (Over.mk g) ⥤ PullbackCone f g }
            {
              obj := fun (c : PullbackCone f g) ↦
                BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst sorry) (Over.homMk c.snd sorry),
              map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦
                { hom := Over.homMk a.hom sorry, w := sorry :
                  ConeMorphism (C := Over.{v, u} X) (F := pair (Over.mk f) (Over.mk g))
                    (BinaryFan.mk (P := Over.mk (c₁.fst ≫ f)) (Over.homMk c₁.fst sorry) (Over.homMk c₁.snd sorry))
                    (BinaryFan.mk (P := Over.mk (c₂.fst ≫ f)) (Over.homMk c₂.fst sorry) (Over.homMk c₂.snd sorry)) },
              map_id := sorry, map_comp := sorry : PullbackCone f g ⥤ BinaryFan (Over.mk f) (Over.mk g) })
          f_1 ≫
        Iso.hom
          ((fun (X_2 : BinaryFan (Over.mk f) (Over.mk g)) ↦
              (BinaryFan.ext
                  (Over.isoMk
                    (Iso.refl
                      (Over.left.{v, u} (X := X)
                        (Cone.pt (J := Discrete WalkingPair) (F := pair (Over.mk f) (Over.mk g))
                          (Functor.obj
                            (Functor.comp (D := PullbackCone f g)
                              {
                                obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
                                  PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) sorry,
                                map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
                                  { hom := Over.Hom.left a.hom, w := sorry },
                                map_id := sorry, map_comp := sorry }
                              {
                                obj := fun (c : PullbackCone f g) ↦
                                  BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst sorry) (Over.homMk c.snd sorry),
                                map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦
                                  { hom := Over.homMk a.hom sorry, w := sorry },
                                map_id := sorry, map_comp := sorry })
                            X_2))))
                    sorry)
                  sorry sorry :
                Functor.obj
                    (Functor.comp (D := PullbackCone f g)
                      {
                        obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
                          PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) sorry,
                        map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
                          { hom := Over.Hom.left a.hom, w := sorry },
                        map_id := sorry, map_comp := sorry }
                      {
                        obj := fun (c : PullbackCone f g) ↦
                          BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst sorry) (Over.homMk c.snd sorry),
                        map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦ { hom := Over.homMk a.hom sorry, w := sorry },
                        map_id := sorry, map_comp := sorry })
                    X_2 ≅
                  Functor.obj (𝟭 (BinaryFan (Over.mk f) (Over.mk g))) X_2))
            Y_1))))
  (Over.Hom.left
    (ConeMorphism.hom
      (Iso.hom
          ((fun (X_2 : BinaryFan (Over.mk f) (Over.mk g)) ↦
              (BinaryFan.ext
                  (Over.isoMk
                    (Iso.refl
                      (Over.left.{v, u} (X := X)
                        (Cone.pt (J := Discrete WalkingPair) (F := pair (Over.mk f) (Over.mk g))
                          (Functor.obj
                            (Functor.comp (D := PullbackCone f g)
                              {
                                obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
                                  PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) sorry,
                                map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
                                  { hom := Over.Hom.left a.hom, w := sorry },
                                map_id := sorry, map_comp := sorry }
                              {
                                obj := fun (c : PullbackCone f g) ↦
                                  BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst sorry) (Over.homMk c.snd sorry),
                                map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦
                                  { hom := Over.homMk a.hom sorry, w := sorry },
                                map_id := sorry, map_comp := sorry })
                            X_2))))
                    sorry)
                  sorry sorry :
                Functor.obj
                    (Functor.comp (D := PullbackCone f g)
                      {
                        obj := fun (c : BinaryFan (Over.mk f) (Over.mk g)) ↦
                          PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) sorry,
                        map := fun {c₁ c₂ : BinaryFan (Over.mk f) (Over.mk g)} (a : c₁ ⟶ c₂) ↦
                          { hom := Over.Hom.left a.hom, w := sorry },
                        map_id := sorry, map_comp := sorry }
                      {
                        obj := fun (c : PullbackCone f g) ↦
                          BinaryFan.mk (P := Over.mk (c.fst ≫ f)) (Over.homMk c.fst sorry) (Over.homMk c.snd sorry),
                        map := fun {c₁ c₂ : PullbackCone f g} (a : c₁ ⟶ c₂) ↦ { hom := Over.homMk a.hom sorry, w := sorry },
                        map_id := sorry, map_comp := sorry })
                    X_2 ≅
                  Functor.obj (𝟭 (BinaryFan (Over.mk f) (Over.mk g))) X_2))
            X_1) ≫
        Functor.map (𝟭 (BinaryFan (Over.mk f) (Over.mk g))) f_1)))
Tactic: ext