-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Limit (goal)
Goal:
J : Type v
inst : SmallCategory J
F : J ⥤ Cat
s : Cone F
X Y : (↑s.1 : Type v)
f : X ⟶ Y
x :
  (j : J) →
    Functor.obj
      (homDiagram (F := F)
        ((hom (FC := TypeCat.Fun)
              (limit.lift (F ⋙ objects)
                { pt := (↑s.pt : Type v),
                  π :=
                    { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ :
                      NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                  Cone (F ⋙ objects) }) :
            (a :
                (fun (X : Type v) ↦ X)
                  { pt := (↑s.pt : Type v),
                      π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }.pt) →
              (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
          X)
        ((hom (FC := TypeCat.Fun)
              (limit.lift (F ⋙ objects)
                { pt := (↑s.pt : Type v),
                  π :=
                    { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ :
                      NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                  Cone (F ⋙ objects) }) :
            (a :
                (fun (X : Type v) ↦ X)
                  { pt := (↑s.pt : Type v),
                      π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }.pt) →
              (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
          Y))
      j
⊢ Quiver.Hom (V := limit (F ⋙ objects))
    ((hom (FC := TypeCat.Fun)
            (limit.lift (F ⋙ objects)
              { pt := (↑s.pt : Type v),
                π :=
                  { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ :
                    NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                Cone (F ⋙ objects) }) :
          (a :
              (fun (X : Type v) ↦ X)
                { pt := (↑s.pt : Type v),
                    π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }.pt) →
            (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
        X :
      (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
    ((hom (FC := TypeCat.Fun)
            (limit.lift (F ⋙ objects)
              { pt := (↑s.pt : Type v),
                π :=
                  { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ :
                    NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                Cone (F ⋙ objects) }) :
          (a :
              (fun (X : Type v) ↦ X)
                { pt := (↑s.pt : Type v),
                    π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }.pt) →
            (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
        Y :
      (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
have h : ∀ (j j' : J) (f_1 : j ⟶ j'),
  Eq (α :=
    (fun (X : Type v) ↦ X)
      (Functor.obj
        (homDiagram (F := F)
          ((hom (FC := TypeCat.Fun)
                (limit.lift (F ⋙ objects)
                  { pt := (↑s.pt : Type v),
                    π :=
                      { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry :
                        NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                    Cone (F ⋙ objects) }) :
              (a :
                  (fun (X : Type v) ↦ X)
                    { pt := (↑s.pt : Type v),
                        π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
            X)
          ((hom (FC := TypeCat.Fun)
                (limit.lift (F ⋙ objects)
                  { pt := (↑s.pt : Type v),
                    π :=
                      { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry :
                        NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                    Cone (F ⋙ objects) }) :
              (a :
                  (fun (X : Type v) ↦ X)
                    { pt := (↑s.pt : Type v),
                        π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
            Y))
        j'))
    ((hom (FC := TypeCat.Fun)
            (Functor.map
              (homDiagram (F := F)
                ((hom (FC := TypeCat.Fun)
                      (limit.lift (F ⋙ objects)
                        { pt := (↑s.pt : Type v),
                          π :=
                            { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry :
                              NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                          Cone (F ⋙ objects) }) :
                    (a :
                        (fun (X : Type v) ↦ X)
                          { pt := (↑s.pt : Type v),
                              π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                      (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
                  X)
                ((hom (FC := TypeCat.Fun)
                      (limit.lift (F ⋙ objects)
                        { pt := (↑s.pt : Type v),
                          π :=
                            { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry :
                              NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                          Cone (F ⋙ objects) }) :
                    (a :
                        (fun (X : Type v) ↦ X)
                          { pt := (↑s.pt : Type v),
                              π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                      (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
                  Y))
              f_1) :
          (a :
              (fun (X : Type v) ↦ X)
                ((homDiagram
                      ((hom
                            (limit.lift (F ⋙ objects)
                              { pt := (↑s.pt : Type v),
                                π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }) :
                          (a :
                              (fun (X : Type v) ↦ X)
                                { pt := (↑s.pt : Type v),
                                    π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                            (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
                        X)
                      ((hom
                            (limit.lift (F ⋙ objects)
                              { pt := (↑s.pt : Type v),
                                π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }) :
                          (a :
                              (fun (X : Type v) ↦ X)
                                { pt := (↑s.pt : Type v),
                                    π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                            (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
                        Y)).obj
                  j)) →
            (fun (X : Type v) ↦ X)
              ((homDiagram
                    ((hom
                          (limit.lift (F ⋙ objects)
                            { pt := (↑s.pt : Type v),
                              π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }) :
                        (a :
                            (fun (X : Type v) ↦ X)
                              { pt := (↑s.pt : Type v),
                                  π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                          (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
                      X)
                    ((hom
                          (limit.lift (F ⋙ objects)
                            { pt := (↑s.pt : Type v),
                              π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }) :
                        (a :
                            (fun (X : Type v) ↦ X)
                              { pt := (↑s.pt : Type v),
                                  π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                          (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
                      Y)).obj
                j'))
        (eqToHom sorry ≫ (s.π.app j).toFunctor.map f ≫ eqToHom sorry) :
      (fun (X : Type v) ↦ X)
        (Functor.obj
          (homDiagram (F := F)
            ((hom (FC := TypeCat.Fun)
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π :=
                        { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry :
                          NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                      Cone (F ⋙ objects) }) :
                (a :
                    (fun (X : Type v) ↦ X)
                      { pt := (↑s.pt : Type v),
                          π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                  (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
              X)
            ((hom (FC := TypeCat.Fun)
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π :=
                        { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry :
                          NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                      Cone (F ⋙ objects) }) :
                (a :
                    (fun (X : Type v) ↦ X)
                      { pt := (↑s.pt : Type v),
                          π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                  (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
              Y))
          j'))
    (eqToHom sorry ≫ (s.π.app j').toFunctor.map f ≫ eqToHom sorry :
      Quiver.Hom (V := (fun (X : Type v) ↦ X) ((F ⋙ objects).obj j'))
        ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') :
              (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj j'))
            ((hom (FC := TypeCat.Fun)
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π :=
                        { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry :
                          NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                      Cone (F ⋙ objects) }) :
                (a :
                    (fun (X : Type v) ↦ X)
                      { pt := (↑s.pt : Type v),
                          π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                  (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
              X) :
          (fun (X : Type v) ↦ X) ((F ⋙ objects).obj j'))
        ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') :
              (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj j'))
            ((hom (FC := TypeCat.Fun)
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π :=
                        { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry :
                          NatTrans (Functor.obj (Functor.const J) (↑s.pt : Type v)) (F ⋙ objects) } :
                      Cone (F ⋙ objects) }) :
                (a :
                    (fun (X : Type v) ↦ X)
                      { pt := (↑s.pt : Type v),
                          π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }.pt) →
                  (fun (X : Type v) ↦ X) (limit (F ⋙ objects)))
              Y) :
          (fun (X : Type v) ↦ X) ((F ⋙ objects).obj j')))
Tactic: fapply Types.Limit.mk.{v, v}