awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Bipointed (goal)
Goal:
X : Pointed
Y : Bipointed
f : pointedToBipointedFst.obj X ⟶ Y
x : (pointedToBipointedFst.obj X).X
⊢ Bipointed.Hom.toFun
      ((fun (f : X ⟶ bipointedToPointedFst.obj Y) ↦
          ({ toFun := fun (o : (pointedToBipointedFst.obj X).X) ↦ Option.elim o Y.toProd.2 f.toFun, map_fst := ⋯,
              map_snd := ⋯ } :
            (pointedToBipointedFst.obj X).Hom Y))
        ((fun (f : pointedToBipointedFst.obj X ⟶ Y) ↦
            { toFun := f.toFun ∘ some, map_point := ⋯ : X.Hom (bipointedToPointedFst.obj Y) })
          f))
      x =
    f.toFun x
have toFun : Bipointed.Hom.toFun (X := pointedToBipointedFst.obj X)
    ((fun (f : X ⟶ bipointedToPointedFst.obj Y) ↦
        ({ toFun := fun (o : (pointedToBipointedFst.obj X).X) ↦ Option.elim o Y.toProd.2 f.toFun, map_fst := sorry,
            map_snd := sorry } :
          (pointedToBipointedFst.obj X).Hom Y))
      ((fun (f : pointedToBipointedFst.obj X ⟶ Y) ↦
          { toFun := f.toFun ∘ some, map_point := sorry : X.Hom (bipointedToPointedFst.obj Y) })
        f))
    none =
  f.toFun none
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Bipointed (goal)
Goal:
X : Pointed
Y : Bipointed
f : pointedToBipointedFst.obj X ⟶ Y
x : (pointedToBipointedFst.obj X).X
toFun :
  Bipointed.Hom.toFun (X := pointedToBipointedFst.obj X)
      ((fun (f : X ⟶ bipointedToPointedFst.obj Y) ↦
          ({ toFun := fun (o : (pointedToBipointedFst.obj X).X) ↦ Option.elim o Y.toProd.2 f.toFun, map_fst := ⋯,
              map_snd := ⋯ } :
            (pointedToBipointedFst.obj X).Hom Y))
        ((fun (f : pointedToBipointedFst.obj X ⟶ Y) ↦
            { toFun := f.toFun ∘ some, map_point := ⋯ : X.Hom (bipointedToPointedFst.obj Y) })
          f))
      none =
    f.toFun none
⊢ Bipointed.Hom.toFun
      ((fun (f : X ⟶ bipointedToPointedFst.obj Y) ↦
          ({ toFun := fun (o : (pointedToBipointedFst.obj X).X) ↦ Option.elim o Y.toProd.2 f.toFun, map_fst := ⋯,
              map_snd := ⋯ } :
            (pointedToBipointedFst.obj X).Hom Y))
        ((fun (f : pointedToBipointedFst.obj X ⟶ Y) ↦
            { toFun := f.toFun ∘ some, map_point := ⋯ : X.Hom (bipointedToPointedFst.obj Y) })
          f))
      x =
    f.toFun x
have toFun₁ : ∀ (val : X.X),
  Bipointed.Hom.toFun (X := pointedToBipointedFst.obj X)
      ((fun (f : X ⟶ bipointedToPointedFst.obj Y) ↦
          ({ toFun := fun (o : (pointedToBipointedFst.obj X).X) ↦ Option.elim o Y.toProd.2 f.toFun, map_fst := sorry,
              map_snd := sorry } :
            (pointedToBipointedFst.obj X).Hom Y))
        ((fun (f : pointedToBipointedFst.obj X ⟶ Y) ↦
            { toFun := f.toFun ∘ some, map_point := sorry : X.Hom (bipointedToPointedFst.obj Y) })
          f))
      (some val) =
    f.toFun (some val)
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Bipointed (goal)
Goal:
X : Pointed
Y : Bipointed
f : pointedToBipointedSnd.obj X ⟶ Y
⊢ (fun (f : X ⟶ bipointedToPointedSnd.obj Y) ↦
        ({ toFun := fun (o : (pointedToBipointedSnd.obj X).X) ↦ Option.elim o Y.toProd.1 f.toFun, map_fst := ⋯,
            map_snd := ⋯ } :
          (pointedToBipointedSnd.obj X).Hom Y))
      ((fun (f : pointedToBipointedSnd.obj X ⟶ Y) ↦
          { toFun := f.toFun ∘ some, map_point := ⋯ : X.Hom (bipointedToPointedSnd.obj Y) })
        f) =
    f
have toFun : Bipointed.Hom.toFun
    ((fun (f : X ⟶ bipointedToPointedSnd.obj Y) ↦
        ({ toFun := fun (o : (pointedToBipointedSnd.obj X).X) ↦ Option.elim o Y.toProd.1 f.toFun, map_fst := sorry,
            map_snd := sorry } :
          (pointedToBipointedSnd.obj X).Hom Y))
      ((fun (f : pointedToBipointedSnd.obj X ⟶ Y) ↦
          { toFun := f.toFun ∘ some, map_point := sorry : X.Hom (bipointedToPointedSnd.obj Y) })
        f)) =
  f.toFun
Tactic: apply Bipointed.Hom.ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Bipointed (goal)
Goal:
X : Pointed
Y : Bipointed
f : pointedToBipointedSnd.obj X ⟶ Y
⊢ Bipointed.Hom.toFun
      ((fun (f : X ⟶ bipointedToPointedSnd.obj Y) ↦
          ({ toFun := fun (o : (pointedToBipointedSnd.obj X).X) ↦ Option.elim o Y.toProd.1 f.toFun, map_fst := ⋯,
              map_snd := ⋯ } :
            (pointedToBipointedSnd.obj X).Hom Y))
        ((fun (f : pointedToBipointedSnd.obj X ⟶ Y) ↦
            { toFun := f.toFun ∘ some, map_point := ⋯ : X.Hom (bipointedToPointedSnd.obj Y) })
          f)) =
    f.toFun
have toFun : ∀ (x : (pointedToBipointedSnd.obj X).X),
  Bipointed.Hom.toFun
      ((fun (f : X ⟶ bipointedToPointedSnd.obj Y) ↦
          ({ toFun := fun (o : (pointedToBipointedSnd.obj X).X) ↦ Option.elim o Y.toProd.1 f.toFun, map_fst := sorry,
              map_snd := sorry } :
            (pointedToBipointedSnd.obj X).Hom Y))
        ((fun (f : pointedToBipointedSnd.obj X ⟶ Y) ↦
            { toFun := f.toFun ∘ some, map_point := sorry : X.Hom (bipointedToPointedSnd.obj Y) })
          f))
      x =
    f.toFun x
Tactic: funext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Bipointed (goal)
Goal:
X : Pointed
Y : Bipointed
f : pointedToBipointedSnd.obj X ⟶ Y
x : (pointedToBipointedSnd.obj X).X
⊢ Bipointed.Hom.toFun
      ((fun (f : X ⟶ bipointedToPointedSnd.obj Y) ↦
          ({ toFun := fun (o : (pointedToBipointedSnd.obj X).X) ↦ Option.elim o Y.toProd.1 f.toFun, map_fst := ⋯,
              map_snd := ⋯ } :
            (pointedToBipointedSnd.obj X).Hom Y))
        ((fun (f : pointedToBipointedSnd.obj X ⟶ Y) ↦
            { toFun := f.toFun ∘ some, map_point := ⋯ : X.Hom (bipointedToPointedSnd.obj Y) })
          f))
      x =
    f.toFun x
have toFun : Bipointed.Hom.toFun (X := pointedToBipointedSnd.obj X)
    ((fun (f : X ⟶ bipointedToPointedSnd.obj Y) ↦
        ({ toFun := fun (o : (pointedToBipointedSnd.obj X).X) ↦ Option.elim o Y.toProd.1 f.toFun, map_fst := sorry,
            map_snd := sorry } :
          (pointedToBipointedSnd.obj X).Hom Y))
      ((fun (f : pointedToBipointedSnd.obj X ⟶ Y) ↦
          { toFun := f.toFun ∘ some, map_point := sorry : X.Hom (bipointedToPointedSnd.obj Y) })
        f))
    none =
  f.toFun none
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Bipointed (goal)
Goal:
X : Pointed
Y : Bipointed
f : pointedToBipointedSnd.obj X ⟶ Y
x : (pointedToBipointedSnd.obj X).X
toFun :
  Bipointed.Hom.toFun (X := pointedToBipointedSnd.obj X)
      ((fun (f : X ⟶ bipointedToPointedSnd.obj Y) ↦
          ({ toFun := fun (o : (pointedToBipointedSnd.obj X).X) ↦ Option.elim o Y.toProd.1 f.toFun, map_fst := ⋯,
              map_snd := ⋯ } :
            (pointedToBipointedSnd.obj X).Hom Y))
        ((fun (f : pointedToBipointedSnd.obj X ⟶ Y) ↦
            { toFun := f.toFun ∘ some, map_point := ⋯ : X.Hom (bipointedToPointedSnd.obj Y) })
          f))
      none =
    f.toFun none
⊢ Bipointed.Hom.toFun
      ((fun (f : X ⟶ bipointedToPointedSnd.obj Y) ↦
          ({ toFun := fun (o : (pointedToBipointedSnd.obj X).X) ↦ Option.elim o Y.toProd.1 f.toFun, map_fst := ⋯,
              map_snd := ⋯ } :
            (pointedToBipointedSnd.obj X).Hom Y))
        ((fun (f : pointedToBipointedSnd.obj X ⟶ Y) ↦
            { toFun := f.toFun ∘ some, map_point := ⋯ : X.Hom (bipointedToPointedSnd.obj Y) })
          f))
      x =
    f.toFun x
have toFun₁ : ∀ (val : X.X),
  Bipointed.Hom.toFun (X := pointedToBipointedSnd.obj X)
      ((fun (f : X ⟶ bipointedToPointedSnd.obj Y) ↦
          ({ toFun := fun (o : (pointedToBipointedSnd.obj X).X) ↦ Option.elim o Y.toProd.1 f.toFun, map_fst := sorry,
              map_snd := sorry } :
            (pointedToBipointedSnd.obj X).Hom Y))
        ((fun (f : pointedToBipointedSnd.obj X ⟶ Y) ↦
            { toFun := f.toFun ∘ some, map_point := sorry : X.Hom (bipointedToPointedSnd.obj Y) })
          f))
      (some val) =
    f.toFun (some val)
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X : Type u
C : Cat
x x_1 : Cat
x_2 : x ⟶ x_1
as : objects.obj x
⊢ ∀ (Y : (↑((objects ⋙ typeToCat).obj x) : Type u_1)) (f : { as := as : Discrete (objects.obj x) } ⟶ Y),
    HEq (((objects ⋙ typeToCat).map x_2 ≫ x_1.typeToCatObjectsAdjCounitApp.toCatHom).toFunctor.map f) (β :=
      Quiver.Hom (V := (↑(Functor.obj (𝟭 Cat) x_1) : Type u_1))
        (Functor.obj (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2))
          { as := as : Discrete (objects.obj x) })
        (Functor.obj (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2)) Y))
      (Functor.map (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2)) f)
have this : ∀ (as_1 : objects.obj x) (f : { as := as : Discrete (objects.obj x) } ⟶ { as := as_1 }),
  HEq (((objects ⋙ typeToCat).map x_2 ≫ x_1.typeToCatObjectsAdjCounitApp.toCatHom).toFunctor.map f) (β :=
    Quiver.Hom (V := (↑(Functor.obj (𝟭 Cat) x_1) : Type u_1))
      (Functor.obj (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2))
        { as := as : Discrete (objects.obj x) })
      (Functor.obj (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2))
        { as := as_1 : Discrete (objects.obj x) }))
    (Functor.map (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2)) f)
Tactic: ⟨_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X : Type u
C : Cat
x x_1 : Cat
x_2 : x ⟶ x_1
as as_1 : objects.obj x
f : { as := as : Discrete (objects.obj x) } ⟶ { as := as_1 }
⊢ HEq (((objects ⋙ typeToCat).map x_2 ≫ x_1.typeToCatObjectsAdjCounitApp.toCatHom).toFunctor.map f) (β :=
    Quiver.Hom (V := (↑(Functor.obj (𝟭 Cat) x_1) : Type u_1))
      (Functor.obj (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2))
        { as := as : Discrete (objects.obj x) })
      (Functor.obj (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2))
        { as := as_1 : Discrete (objects.obj x) }))
    (Functor.map (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2)) f)
have obtain : ∀ (f : { as := as : Discrete (objects.obj x) } ⟶ { as := { as := as }.as }),
  HEq (((objects ⋙ typeToCat).map x_2 ≫ x_1.typeToCatObjectsAdjCounitApp.toCatHom).toFunctor.map f) (β :=
    Quiver.Hom (V := (↑(Functor.obj (𝟭 Cat) x_1) : Type u_1))
      (Functor.obj (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2))
        { as := as : Discrete (objects.obj x) })
      (Functor.obj (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2))
        { as := { as := as : Discrete (objects.obj x) }.as : Discrete (objects.obj x) }))
    (Functor.map (Hom.toFunctor (x.typeToCatObjectsAdjCounitApp.toCatHom ≫ Functor.map (𝟭 Cat) x_2)) f)
Tactic: obtain rfl := Discrete.eq_of_hom f
Cleared hypotheses: as_1, f
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X : Type u
C : Cat
x : Cat
⊢ ↾(Hom.toFunctor (𝟙 x)).mapConnectedComponents = 𝟙 (ConnectedComponents.{v, u} (↑x : Type u))
have h : ∀ (x_1 : ConnectedComponents.{v, u} (↑x : Type u)),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (↾(Hom.toFunctor (𝟙 x)).mapConnectedComponents)).toFun x_1 =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 (ConnectedComponents.{v, u} (↑x : Type u)))).toFun x_1
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X : Type u
C : Cat
x_1 : Cat
x : ConnectedComponents.{v, u} (↑x_1 : Type u)
x_2 : (↑x_1 : Type u)
h : ⟦x_2⟧ = x
⊢ Functor.mapConnectedComponents (𝟭 (↑x_1 : Type u)) x = x
have subst : Eq (α := ConnectedComponents.{v, u} (↑x_1 : Type u)) (Functor.mapConnectedComponents (𝟭 (↑x_1 : Type u)) ⟦x_2⟧) ⟦x_2⟧
Tactic: subst h
Cleared hypotheses: x, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X : Type u
C : Cat
X_1 Y Z : Cat
x : X_1 ⟶ Y
x_1 : Y ⟶ Z
⊢ ↾(x ≫ x_1).toFunctor.mapConnectedComponents =
    ↾x.toFunctor.mapConnectedComponents ≫ ↾x_1.toFunctor.mapConnectedComponents
have h : ∀ (x_2 : ConnectedComponents.{v, u} (↑X_1 : Type u)),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (↾(x ≫ x_1).toFunctor.mapConnectedComponents)).toFun x_2 =
    (ConcreteCategory.hom (FC := TypeCat.Fun)
          (↾x.toFunctor.mapConnectedComponents ≫ ↾x_1.toFunctor.mapConnectedComponents)).toFun
      x_2
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X : Type u
C : Cat
X_1 Y Z : Cat
x_1 : X_1 ⟶ Y
x_2 : Y ⟶ Z
x : ConnectedComponents.{v, u} (↑X_1 : Type u)
x_3 : (↑X_1 : Type u)
h : ⟦x_3⟧ = x
⊢ (x_1.toFunctor ⋙ x_2.toFunctor).mapConnectedComponents x =
    x_2.toFunctor.mapConnectedComponents (x_1.toFunctor.mapConnectedComponents x)
have subst : (x_1.toFunctor ⋙ x_2.toFunctor).mapConnectedComponents ⟦x_3⟧ =
  x_2.toFunctor.mapConnectedComponents (x_1.toFunctor.mapConnectedComponents ⟦x_3⟧)
Tactic: subst h
Cleared hypotheses: x, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X_1 : Type u
C_1 : Cat
C : Cat
X : Type u
⊢ (connectedComponents.obj C ⟶ X) ≃ (C ⟶ typeToCat.obj X)
have refine : (connectedComponents.obj C → X) ≃ (C ⟶ typeToCat.obj X)
Tactic: refine TypeCat.homEquiv.trans ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X : Type u
C : Cat
x x_1 : Type u
x_2 : x ⟶ x_1
⊢ Eq (α := (typeToCat ⋙ connectedComponents).obj x ⟶ Functor.obj (𝟭 (Type u)) x_1)
    ((typeToCat ⋙ connectedComponents).map x_2 ≫
      ↾ConnectedComponents.liftFunctor (↑(typeToCat.obj x_1) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj x_1)))
    (↾ConnectedComponents.liftFunctor (↑(typeToCat.obj x) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj x)) ≫
      Functor.map (𝟭 (Type u)) x_2)
have h : ∀ (xcc : (typeToCat ⋙ connectedComponents).obj x),
  Eq (α := Functor.obj (𝟭 (Type u)) x_1)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((typeToCat ⋙ connectedComponents).map x_2 ≫
          ↾ConnectedComponents.liftFunctor (↑(typeToCat.obj x_1) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj x_1))))
      xcc)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (↾ConnectedComponents.liftFunctor (↑(typeToCat.obj x) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj x)) ≫
          Functor.map (𝟭 (Type u)) x_2))
      xcc)
Tactic: ext xcc
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X : Type u
C : Cat
x x_1 : Type u
x_2 : x ⟶ x_1
xcc : (typeToCat ⋙ connectedComponents).obj x
⊢ Eq (α := Functor.obj (𝟭 (Type u)) x_1)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((typeToCat ⋙ connectedComponents).map x_2 ≫
          ↾ConnectedComponents.liftFunctor (↑(typeToCat.obj x_1) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj x_1))))
      xcc)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (↾ConnectedComponents.liftFunctor (↑(typeToCat.obj x) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj x)) ≫
          Functor.map (𝟭 (Type u)) x_2))
      xcc)
have h : ∀ (x_3 : (↑(typeToCat.obj x) : Type u)) (h : ⟦x_3⟧ = xcc),
  Eq (α := Functor.obj (𝟭 (Type u)) x_1)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((typeToCat ⋙ connectedComponents).map x_2 ≫
          ↾ConnectedComponents.liftFunctor (↑(typeToCat.obj x_1) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj x_1))))
      xcc)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (↾ConnectedComponents.liftFunctor (↑(typeToCat.obj x) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj x)) ≫
          Functor.map (𝟭 (Type u)) x_2))
      xcc)
Tactic: obtain ⟨x, h⟩ := Quotient.exists_rep xcc
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X_1 : Type u
C_1 : Cat
C : Cat
X : Type u
G : C ⟶ typeToCat.obj X
⊢ Eq (α := connectedComponents.obj C ⟶ X)
    ((Equiv.symm
            (Equiv.trans TypeCat.homEquiv
              (Equiv.trans (ConnectedComponents.typeToCatHomEquiv (↑C : Type u) X)
                (Functor.equivCatHom (↑C : Type u) (Discrete X)))) :
          (a : C ⟶ typeToCat.obj X) → connectedComponents.obj C ⟶ X)
        G :
      connectedComponents.obj C ⟶ X)
    (connectedComponents.map G ≫
        NatTrans.app
          {
            app := fun (X : Type u) ↦
              ↾ConnectedComponents.liftFunctor (↑(typeToCat.obj X) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj X)),
            naturality := ⋯ }
          X :
      connectedComponents.obj C ⟶ Functor.obj (𝟭 (Type u)) X)
have h : ∀ (cc : connectedComponents.obj C),
  Eq (α := X)
    (TypeCat.Fun.toFun
      (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
        ((Equiv.symm
              (Equiv.trans TypeCat.homEquiv
                (Equiv.trans (ConnectedComponents.typeToCatHomEquiv (↑C : Type u) X)
                  (Functor.equivCatHom (↑C : Type u) (Discrete X)))) :
            (a : C ⟶ typeToCat.obj X) → connectedComponents.obj C ⟶ X)
          G))
      cc)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (connectedComponents.map G ≫
          NatTrans.app
            {
              app := fun (X : Type u) ↦
                ↾ConnectedComponents.liftFunctor (↑(typeToCat.obj X) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj X)),
              naturality := sorry : NatTrans (typeToCat ⋙ connectedComponents) (𝟭 (Type u)) }
            X))
      cc)
Tactic: ext cc
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Cat.Adjunction (goal)
Goal:
X_1 : Type u
C_1 : Cat
C : Cat
X : Type u
G : C ⟶ typeToCat.obj X
cc : connectedComponents.obj C
⊢ Eq (α := X)
    (TypeCat.Fun.toFun
      (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
        ((Equiv.symm
              (Equiv.trans TypeCat.homEquiv
                (Equiv.trans (ConnectedComponents.typeToCatHomEquiv (↑C : Type u) X)
                  (Functor.equivCatHom (↑C : Type u) (Discrete X)))) :
            (a : C ⟶ typeToCat.obj X) → connectedComponents.obj C ⟶ X)
          G))
      cc)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (connectedComponents.map G ≫
          NatTrans.app
            {
              app := fun (X : Type u) ↦
                ↾ConnectedComponents.liftFunctor (↑(typeToCat.obj X) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj X)),
              naturality := ⋯ : NatTrans (typeToCat ⋙ connectedComponents) (𝟭 (Type u)) }
            X))
      cc)
have h : ∀ (w : (↑C : Type u)),
  ⟦w⟧ = cc →
    Eq (α := X)
      (TypeCat.Fun.toFun
        (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
          ((Equiv.symm
                (Equiv.trans TypeCat.homEquiv
                  (Equiv.trans (ConnectedComponents.typeToCatHomEquiv (↑C : Type u) X)
                    (Functor.equivCatHom (↑C : Type u) (Discrete X)))) :
              (a : C ⟶ typeToCat.obj X) → connectedComponents.obj C ⟶ X)
            G))
        cc)
      (TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (connectedComponents.map G ≫
            NatTrans.app
              {
                app := fun (X : Type u) ↦
                  ↾ConnectedComponents.liftFunctor (↑(typeToCat.obj X) : Type u) (Hom.toFunctor (𝟙 typeToCat.obj X)),
                naturality := sorry : NatTrans (typeToCat ⋙ connectedComponents) (𝟭 (Type u)) }
              X))
        cc)
Tactic: obtain ⟨_, _⟩ := Quotient.exists_rep cc
-- 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_1 Y_1 : J
f : X_1 ⟶ Y_1
g :
  Quiver.Hom (V := (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) X_1) :
          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
        X :
      (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) X_1) :
          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
        Y :
      (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
⊢ Quiver.Hom (V := (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) Y_1) :
          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
        X :
      (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) Y_1) :
          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
        Y :
      (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
have refine_1 : (hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) Y_1) :
      (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
    X =
  (F.map f).toFunctor.obj
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) X_1) :
        (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
      X)
Tactic: refine eqToHom ?_ ≫ (F.map f).toFunctor.map g ≫ eqToHom ?_
-- 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_1 Y_1 : J
f : X_1 ⟶ Y_1
g :
  Quiver.Hom (V := (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) X_1) :
          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
        X :
      (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) X_1) :
          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
        Y :
      (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
refine_1 :
  (hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) Y_1) :
        (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
      X =
    (F.map f).toFunctor.obj
      ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) X_1) :
          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
        X)
⊢ Quiver.Hom (V := (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) Y_1) :
          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
        X :
      (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) Y_1) :
          (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
        Y :
      (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
have refine_2 : (F.map f).toFunctor.obj
    ((hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) X_1) :
        (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj X_1))
      Y) =
  (hom (FC := TypeCat.Fun) (limit.π (F ⋙ objects) Y_1) :
      (a : (fun (X : Type v) ↦ X) (limit (F ⋙ objects))) → (fun (X : Type v) ↦ X) ((F ⋙ objects).obj Y_1))
    Y
Tactic: refine eqToHom ?_ ≫ (F.map f).toFunctor.map g ≫ eqToHom ?_