awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Limit (goal)
Goal:
J : Type v
inst : SmallCategory J
F : J ⥤ Cat
X Y : limit (F ⋙ objects)
x : limit (homDiagram X Y)
⊢ Types.Limit.mk (homDiagram X Y)
      (fun (j : J) ↦
        CategoryStruct.comp (Y :=
          (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))
            X)
          ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X X) j) :
              (a : (fun (X : Type v) ↦ X) (limit (homDiagram X X))) → (fun (X : Type v) ↦ X) ((homDiagram X X).obj j))
            (Types.Limit.mk (homDiagram X X)
              (fun (x : J) ↦
                𝟙
                  ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) x) :
                      (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj x))
                    X))
              ⋯))
          ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X Y) j) :
              (a : (fun (X : Type v) ↦ X) (limit (homDiagram X Y))) → (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
            x))
      ⋯ =
    x
have w : ∀ (j : J),
  Eq (α := (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
    ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X Y) j) :
          (a : (fun (X : Type v) ↦ X) (limit (homDiagram X Y))) → (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
        (Types.Limit.mk (homDiagram X Y)
          (fun (j : J) ↦
            CategoryStruct.comp (Y :=
              (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))
                X)
              ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X X) j) :
                  (a : (fun (X : Type v) ↦ X) (limit (homDiagram X X))) →
                    (fun (X : Type v) ↦ X) ((homDiagram X X).obj j))
                (Types.Limit.mk (homDiagram X X)
                  (fun (x : J) ↦
                    𝟙
                      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) x) :
                          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) →
                            (fun (X : Type v) ↦ X) ((F ⋙ objects).obj x))
                        X))
                  sorry))
              ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X Y) j) :
                  (a : (fun (X : Type v) ↦ X) (limit (homDiagram X Y))) →
                    (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
                x))
          sorry) :
      (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
    ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X Y) j) :
          (a : (fun (X : Type v) ↦ X) (limit (homDiagram X Y))) → (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
        x :
      (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
Tactic: apply Types.limit_ext.{v, v}
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Limit (goal)
Goal:
J : Type v
inst : SmallCategory J
F : J ⥤ Cat
X Y : limit (F ⋙ objects)
x : limit (homDiagram X Y)
⊢ Types.Limit.mk (homDiagram X Y)
      (fun (j : J) ↦
        CategoryStruct.comp (Y :=
          (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))
            Y)
          ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X Y) j) :
              (a : (fun (X : Type v) ↦ X) (limit (homDiagram X Y))) → (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
            x)
          ((hom (FC := TypeCat.Fun) (limit.π (homDiagram Y Y) j) :
              (a : (fun (X : Type v) ↦ X) (limit (homDiagram Y Y))) → (fun (X : Type v) ↦ X) ((homDiagram Y Y).obj j))
            (Types.Limit.mk (homDiagram Y Y)
              (fun (x : J) ↦
                𝟙
                  ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) x) :
                      (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj x))
                    Y))
              ⋯)))
      ⋯ =
    x
have w : ∀ (j : J),
  Eq (α := (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
    ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X Y) j) :
          (a : (fun (X : Type v) ↦ X) (limit (homDiagram X Y))) → (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
        (Types.Limit.mk (homDiagram X Y)
          (fun (j : J) ↦
            CategoryStruct.comp (Y :=
              (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))
                Y)
              ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X Y) j) :
                  (a : (fun (X : Type v) ↦ X) (limit (homDiagram X Y))) →
                    (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
                x)
              ((hom (FC := TypeCat.Fun) (limit.π (homDiagram Y Y) j) :
                  (a : (fun (X : Type v) ↦ X) (limit (homDiagram Y Y))) →
                    (fun (X : Type v) ↦ X) ((homDiagram Y Y).obj j))
                (Types.Limit.mk (homDiagram Y Y)
                  (fun (x : J) ↦
                    𝟙
                      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) x) :
                          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) →
                            (fun (X : Type v) ↦ X) ((F ⋙ objects).obj x))
                        Y))
                  sorry)))
          sorry) :
      (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
    ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X Y) j) :
          (a : (fun (X : Type v) ↦ X) (limit (homDiagram X Y))) → (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
        x :
      (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
Tactic: apply Types.limit_ext.{v, v}
-- 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
⊢ 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 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 := 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
Tactic: fapply Types.Limit.mk.{v, v}
-- 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
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
have x : (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) =
  (s.π.app j).toFunctor.obj X
Tactic: refine eqToHom ?_ ≫ (s.π.app j).toFunctor.map f ≫ eqToHom ?_
-- 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
j : J
x :
  (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 := ⋯ :
                    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) =
    (s.π.app j).toFunctor.obj 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 := ⋯ :
                    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
have x₁ : (s.π.app j).toFunctor.obj Y =
  (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)
Tactic: refine eqToHom ?_ ≫ (s.π.app j).toFunctor.map f ≫ eqToHom ?_
-- 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
j j' : J
h : j ⟶ j'
⊢ Eq (α :=
    Quiver.Hom (V := objects.obj (F.obj j'))
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
          ((@hom _ types TypeCat.Fun _ _ _ _ _
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }) :
                (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
              X :
            limit (F ⋙ objects)) :
        objects.obj (F.obj j'))
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
          ((@hom _ types TypeCat.Fun _ _ _ _ _
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }) :
                (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
              Y :
            limit (F ⋙ objects)) :
        objects.obj (F.obj j')))
    (eqToHom ⋯ ≫ (F.map h).toFunctor.map (eqToHom ⋯ ≫ (s.π.app j).toFunctor.map f ≫ eqToHom ⋯) ≫ eqToHom ⋯)
    (eqToHom ⋯ ≫ (s.π.app j').toFunctor.map f ≫ eqToHom ⋯)
have h₁ : Eq (α :=
  Quiver.Hom (V := objects.obj (F.obj j'))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
        ((@hom _ types TypeCat.Fun _ _ _ _ _
                (limit.lift (F ⋙ objects)
                  { pt := (↑s.pt : Type v),
                    π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }) :
              (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
            X :
          limit (F ⋙ objects)) :
      objects.obj (F.obj j'))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
        ((@hom _ types TypeCat.Fun _ _ _ _ _
                (limit.lift (F ⋙ objects)
                  { pt := (↑s.pt : Type v),
                    π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }) :
              (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
            Y :
          limit (F ⋙ objects)) :
      objects.obj (F.obj j')))
  (eqToHom sorry ≫ ((s.π.app j).toFunctor ⋙ (F.map h).toFunctor).map f ≫ eqToHom sorry)
  (eqToHom sorry ≫ (s.π.app j').toFunctor.map f ≫ eqToHom sorry)
Tactic: simp only [Functor.map_comp, eqToHom_map, ← Functor.comp_map, Category.assoc, eqToHom_trans,
          eqToHom_trans_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Limit (hypothesis)
Goal:
J : Type v
inst : SmallCategory J
F : J ⥤ Cat
s : Cone F
X Y : (↑s.1 : Type v)
f : X ⟶ Y
j j' : J
h : j ⟶ j'
⊢ Eq (α :=
    Quiver.Hom (V := objects.obj (F.obj j'))
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
          ((@hom _ types TypeCat.Fun _ _ _ _ _
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }) :
                (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
              X :
            limit (F ⋙ objects)) :
        objects.obj (F.obj j'))
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
          ((@hom _ types TypeCat.Fun _ _ _ _ _
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }) :
                (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
              Y :
            limit (F ⋙ objects)) :
        objects.obj (F.obj j')))
    (eqToHom ⋯ ≫ ((s.π.app j).toFunctor ⋙ (F.map h).toFunctor).map f ≫ eqToHom ⋯)
    (eqToHom ⋯ ≫ (s.π.app j').toFunctor.map f ≫ eqToHom ⋯)
have this : (s.π.app j ≫ F.map h).toFunctor = (Functor.map (Functor.obj (Functor.const J) s.pt) h ≫ s.π.app j').toFunctor
Tactic: have := congr($((s.π.naturality h).symm).toFunctor)
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Limit (hypothesis)
Goal:
J : Type v
inst : SmallCategory J
F : J ⥤ Cat
s : Cone F
X Y : (↑s.1 : Type v)
f : X ⟶ Y
j j' : J
h : j ⟶ j'
this : (s.π.app j).toFunctor ⋙ (F.map h).toFunctor = 𝟭 (↑s.pt : Type v) ⋙ (s.π.app j').toFunctor
⊢ Eq (α :=
    Quiver.Hom (V := objects.obj (F.obj j'))
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
          ((@hom _ types TypeCat.Fun _ _ _ _ _
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }) :
                (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
              X :
            limit (F ⋙ objects)) :
        objects.obj (F.obj j'))
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
          ((@hom _ types TypeCat.Fun _ _ _ _ _
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }) :
                (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
              Y :
            limit (F ⋙ objects)) :
        objects.obj (F.obj j')))
    (eqToHom ⋯ ≫ ((s.π.app j).toFunctor ⋙ (F.map h).toFunctor).map f ≫ eqToHom ⋯)
    (eqToHom ⋯ ≫ (s.π.app j').toFunctor.map f ≫ eqToHom ⋯)
have this₁ : (s.π.app j).toFunctor ⋙ (F.map h).toFunctor = (s.π.app j').toFunctor
Tactic: rw [Functor.id_comp] at this
Cleared hypotheses: this
-- 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
j j' : J
h : j ⟶ j'
this : (s.π.app j).toFunctor ⋙ (F.map h).toFunctor = (s.π.app j').toFunctor
⊢ Eq (α :=
    Quiver.Hom (V := objects.obj (F.obj j'))
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
          ((@hom _ types TypeCat.Fun _ _ _ _ _
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }) :
                (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
              X :
            limit (F ⋙ objects)) :
        objects.obj (F.obj j'))
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
          ((@hom _ types TypeCat.Fun _ _ _ _ _
                  (limit.lift (F ⋙ objects)
                    { pt := (↑s.pt : Type v),
                      π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := ⋯ } }) :
                (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
              Y :
            limit (F ⋙ objects)) :
        objects.obj (F.obj j')))
    (eqToHom ⋯ ≫ ((s.π.app j).toFunctor ⋙ (F.map h).toFunctor).map f ≫ eqToHom ⋯)
    (eqToHom ⋯ ≫ (s.π.app j').toFunctor.map f ≫ eqToHom ⋯)
have h₁ : Eq (α :=
  Quiver.Hom (V := objects.obj (F.obj j'))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
        ((@hom _ types TypeCat.Fun _ _ _ _ _
                (limit.lift (F ⋙ objects)
                  { pt := (↑s.pt : Type v),
                    π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }) :
              (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
            X :
          limit (F ⋙ objects)) :
      objects.obj (F.obj j'))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j') : (a : limit (F ⋙ objects)) → objects.obj (F.obj j'))
        ((@hom _ types TypeCat.Fun _ _ _ _ _
                (limit.lift (F ⋙ objects)
                  { pt := (↑s.pt : Type v),
                    π := { app := fun (j : J) ↦ ↾(s.π.app j).toFunctor.obj, naturality := sorry } }) :
              (a : (↑s.pt : Type v)) → limit (F ⋙ objects))
            Y :
          limit (F ⋙ objects)) :
      objects.obj (F.obj j')))
  (eqToHom sorry ≫ (eqToHom sorry ≫ (s.π.app j').toFunctor.map f ≫ eqToHom sorry) ≫ eqToHom sorry)
  (eqToHom sorry ≫ (s.π.app j').toFunctor.map f ≫ eqToHom sorry)
Tactic: rw [Functor.congr_hom this f]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Limit (goal)
Goal:
J : Type v
inst : SmallCategory J
F : J ⥤ Cat
X Y : limit (F ⋙ objects)
j : J
h : X = Y
⊢ Eq (α := (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
    ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X Y) j) :
          (a : (fun (X : Type v) ↦ X) (limit (homDiagram X Y))) → (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
        (eqToHom h) :
      (fun (X : Type v) ↦ X) ((homDiagram X Y).obj j))
    (eqToHom ⋯ :
      Quiver.Hom (V := (F ⋙ objects).obj j)
        ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j) : (a : limit (F ⋙ objects)) → (F ⋙ objects).obj j) X :
          (F ⋙ objects).obj j)
        ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j) : (a : limit (F ⋙ objects)) → (F ⋙ objects).obj j) Y :
          (F ⋙ objects).obj j))
have subst : Eq (α := (fun (X : Type v) ↦ X) ((homDiagram X X).obj j))
  ((hom (FC := TypeCat.Fun) (limit.π (homDiagram X X) j) :
        (a : (fun (X : Type v) ↦ X) (limit (homDiagram X X))) → (fun (X : Type v) ↦ X) ((homDiagram X X).obj j))
      (eqToHom sorry) :
    (fun (X : Type v) ↦ X) ((homDiagram X X).obj j))
  (eqToHom sorry :
    Quiver.Hom (V := (F ⋙ objects).obj j)
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j) : (a : limit (F ⋙ objects)) → (F ⋙ objects).obj j) X :
        (F ⋙ objects).obj j)
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) j) : (a : limit (F ⋙ objects)) → (F ⋙ objects).obj j) X :
        (F ⋙ objects).obj j))
Tactic: subst h
Cleared hypotheses: Y, h