awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Category.Compactum
Goal:
e : forget CompHaus (FC := fun (x1 x2 : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
    C((↑(x1.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1),
      (↑(x2.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))
    (CC := fun (X : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
    (↑(X.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)) ≅
  compactumToCompHaus.inv ⋙ Compactum.forget :=
  ((Iso.symm
          (Functor.leftUnitor
            (forget CompHaus (FC := fun (x1 x2 : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
              C((↑(x1.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1),
                (↑(x2.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))
              (CC := fun (X : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
              (↑(X.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))) ≪≫
        Functor.isoWhiskerRight compactumToCompHaus.asEquivalence.symm.unitIso
          (forget CompHaus (FC := fun (x1 x2 : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
            C((↑(x1.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1),
              (↑(x2.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))
            (CC := fun (X : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
            (↑(X.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))) ≪≫
      compactumToCompHaus.inv.associator compactumToCompHaus
        (forget CompHaus (FC := fun (x1 x2 : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
          C((↑(x1.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1),
            (↑(x2.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))
          (CC := fun (X : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
          (↑(X.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))) ≪≫
    compactumToCompHaus.inv.isoWhiskerLeft compactumToCompHausCompForget
⊢ CreatesLimits
    (forget CompHaus (FC := fun (x1 x2 : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
      C((↑(x1.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1),
        (↑(x2.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))
      (CC := fun (X : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
      (↑(X.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))

Tactic:
exact createsLimitsOfNatIso e.symm

Premises used: CategoryTheory.Iso.symm, CategoryTheory.createsLimitsOfNatIso
-- Module: Mathlib.Topology.Category.Compactum
Goal:
⊢ CreatesLimits
    (profiniteToCompHaus ⋙
      forget CompHaus (FC := fun (x1 x2 : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
        C((↑(x1.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1),
          (↑(x2.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))
        (CC := fun (X : CompHausLike fun (x : TopCat.{u_1}) ↦ True) ↦
        (↑(X.toTop (P := fun (x : TopCat.{u_1}) ↦ True)) : Type u_1)))

Tactic:
infer_instance
-- Module: Mathlib.Topology.Category.DeltaGenerated
Goal:
⊢ deltaGeneratedToTop.Full

Tactic:
fullyFaithfulDeltaGeneratedToTop.full

Premises used: DeltaGenerated.fullyFaithfulDeltaGeneratedToTop, CategoryTheory.Functor.FullyFaithful.full
-- Module: Mathlib.Topology.Category.DeltaGenerated
Goal:
⊢ deltaGeneratedToTop.Faithful

Tactic:
fullyFaithfulDeltaGeneratedToTop.faithful

Premises used: DeltaGenerated.fullyFaithfulDeltaGeneratedToTop, CategoryTheory.Functor.FullyFaithful.faithful
-- Module: Mathlib.Topology.Category.DeltaGenerated
Goal:
X Y : TopCat
a₁ a₂ : X ⟶ Y
h : topToDeltaGenerated.map a₁ = topToDeltaGenerated.map a₂
w :
  ∀ (x : (↑X : Type u)),
    Eq (α := (↑Y : Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              a₁ :
            (a : (↑X : Type u)) → (↑Y : Type u))
          x :
        (↑Y : Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              a₂ :
            (a : (↑X : Type u)) → (↑Y : Type u))
          x :
        (↑Y : Type u))
⊢ a₁ = a₂

Tactic:
ext x
-- Module: Mathlib.Topology.Category.DeltaGenerated
Goal:
X Y : TopCat
a₁ a₂ : X ⟶ Y
h : topToDeltaGenerated.map a₁ = topToDeltaGenerated.map a₂
x : (↑X : Type u)
⊢ Eq (α := (↑Y : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) a₁ :
          (a : (↑X : Type u)) → (↑Y : Type u))
        x :
      (↑Y : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) a₂ :
          (a : (↑X : Type u)) → (↑Y : Type u))
        x :
      (↑Y : Type u))

Tactic:
CategoryTheory.congr_fun h x

Premises used: Set.Mem, iSup, setOf, CategoryTheory.congr_fun, DeltaGeneratedSpace.of
-- Module: Mathlib.Topology.Category.DeltaGenerated
Goal:
X Y : TopCat
a₁ a₂ : X ⟶ Y
h : topToDeltaGenerated.map a₁ = topToDeltaGenerated.map a₂
x : (↑X : Type u)
⊢ Eq (α := (↑Y : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) a₁ :
          (a : (↑X : Type u)) → (↑Y : Type u))
        x :
      (↑Y : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) a₂ :
          (a : (↑X : Type u)) → (↑Y : Type u))
        x :
      (↑Y : Type u))

Tactic:
exact CategoryTheory.congr_fun h x

Premises used: CategoryTheory.congr_fun
-- Module: Mathlib.Topology.Category.DeltaGenerated
Goal:
⊢ Limits.HasLimits DeltaGenerated

Tactic:
hasLimits_of_coreflective deltaGeneratedToTop

Premises used: DeltaGenerated.deltaGeneratedToTop, CategoryTheory.hasLimits_of_coreflective
-- Module: Mathlib.Topology.Category.DeltaGenerated
Goal:
⊢ Limits.HasColimits DeltaGenerated

Tactic:
hasColimits_of_hasColimits_createsColimits deltaGeneratedToTop

Premises used: DeltaGenerated.deltaGeneratedToTop, CategoryTheory.hasColimits_of_hasColimits_createsColimits
-- Module: Mathlib.Topology.Category.FinTopCat
Goal:
X : Type u
inst : Fintype X
inst_1 : TopologicalSpace X
⊢ (↑(of X).toTop : Type u) = X

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.Topology.Category.FinTopCat
Goal:
X : FintypeCat
⊢ DiscreteTopology.{u_1} X.obj

Tactic:
⟨rfl⟩

Premises used: DiscreteTopology.mk, rfl
-- Module: Mathlib.Topology.Category.LightProfinite.AsLimit
Goal:
S : LightProfinite
x x_1 : ℕᵒᵖ
x_2 : x ⟶ x_1
simp :
  CategoryTheory.Functor.map (Functor.obj (Functor.const ℕᵒᵖ) S) x_2 ≫
      (lightToProfiniteFullyFaithful.preimageIso
            ((Cone.forget (S.diagram ⋙ lightToProfinite)).mapIso S.isoMapCone)).inv ≫
        S.asLimitConeAux.π.app x_1 =
    (lightToProfiniteFullyFaithful.preimageIso ((Cone.forget (S.diagram ⋙ lightToProfinite)).mapIso S.isoMapCone)).inv ≫
      S.asLimitConeAux.π.app x_1
⊢ CategoryTheory.Functor.map (Functor.obj (Functor.const ℕᵒᵖ) S) x_2 ≫
      (lightToProfiniteFullyFaithful.preimageIso
            ((Cone.forget (S.diagram ⋙ lightToProfinite)).mapIso S.isoMapCone)).inv ≫
        S.asLimitConeAux.π.app x_1 =
    ((lightToProfiniteFullyFaithful.preimageIso
            ((Cone.forget (S.diagram ⋙ lightToProfinite)).mapIso S.isoMapCone)).inv ≫
        S.asLimitConeAux.π.app x) ≫
      S.diagram.map x_2

Tactic:
simp only [Category.assoc, S.asLimitConeAux.w]

Premises used: CategoryTheory.Category.assoc, LightProfinite.asLimitConeAux, CategoryTheory.Limits.Cone.w
-- Module: Mathlib.Topology.Category.LightProfinite.AsLimit
Goal:
S : LightProfinite
x x_1 : ℕᵒᵖ
x_2 : x ⟶ x_1
⊢ CategoryTheory.Functor.map (Functor.obj (Functor.const ℕᵒᵖ) S) x_2 ≫
      (lightToProfiniteFullyFaithful.preimageIso
            ((Cone.forget (S.diagram ⋙ lightToProfinite)).mapIso S.isoMapCone)).inv ≫
        S.asLimitConeAux.π.app x_1 =
    (lightToProfiniteFullyFaithful.preimageIso ((Cone.forget (S.diagram ⋙ lightToProfinite)).mapIso S.isoMapCone)).inv ≫
      S.asLimitConeAux.π.app x_1

Tactic:
rfl
-- Module: Mathlib.Topology.Category.LightProfinite.AsLimit
Goal:
S : LightProfinite
x : ℕᵒᵖ
rw :
  (lightToProfiniteFullyFaithful.preimageIso ((Cone.forget (S.diagram ⋙ lightToProfinite)).mapIso S.isoMapCone)).inv ≫
      S.asLimitConeAux.π.app x =
    S.asLimitCone.π.app x
⊢ S.asLimitConeAux.π.app x =
    (lightToProfiniteFullyFaithful.preimageIso ((Cone.forget (S.diagram ⋙ lightToProfinite)).mapIso S.isoMapCone)).hom ≫
      S.asLimitCone.π.app x

Tactic:
rw [← @Iso.inv_comp_eq]

Premises used: CategoryTheory.Iso.inv_comp_eq
-- Module: Mathlib.Topology.Category.LightProfinite.AsLimit
Goal:
S : LightProfinite
x : ℕᵒᵖ
⊢ (lightToProfiniteFullyFaithful.preimageIso ((Cone.forget (S.diagram ⋙ lightToProfinite)).mapIso S.isoMapCone)).inv ≫
      S.asLimitConeAux.π.app x =
    S.asLimitCone.π.app x

Tactic:
rfl
-- Module: Mathlib.Topology.Category.LightProfinite.AsLimit
Goal:
S : LightProfinite
n : ℕ
⊢ S.proj (n + 1) ≫ S.diagram.map (Opposite.op (homOfLE ⋯)) = S.proj n

Tactic:
S.asLimitCone.w (homOfLE (Nat.le_succ n)).op

Premises used: Quiver.Hom.op, Nat.le_succ, CategoryTheory.Limits.Cone.w, CategoryTheory.homOfLE, LightProfinite.asLimitCone
-- Module: Mathlib.Topology.Category.LightProfinite.AsLimit
Goal:
S : LightProfinite
n m : ℕ
h : n ≤ m
⊢ S.proj m ≫ S.diagram.map (Opposite.op (homOfLE h)) = S.proj n

Tactic:
S.asLimitCone.w (homOfLE h).op

Premises used: Quiver.Hom.op, CategoryTheory.Limits.Cone.w, CategoryTheory.homOfLE, LightProfinite.asLimitCone
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : TotallyDisconnectedSpace X
inst_2 : SecondCountableTopology X
⊢ HasProp
    (fun (Y : TopCat.{u_1}) ↦
      TotallyDisconnectedSpace.{u_1} (↑Y : Type u_1) ∧ SecondCountableTopology.{u_1} (↑Y : Type u_1))
    X

Tactic:
⟨⟨(inferInstance : TotallyDisconnectedSpace X), (inferInstance : SecondCountableTopology X)⟩⟩

Premises used: inferInstance, And.intro, CompHausLike.HasProp.mk
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
X : LightProfinite
⊢ TotallyDisconnectedSpace.{u_1}
    (↑(X.toTop (P := fun (X : TopCat.{u_1}) ↦
          TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
      Type u_1)

Tactic:
X.prop.1

Premises used: And.left, CompHausLike.prop
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
X : LightProfinite
⊢ SecondCountableTopology.{u_1}
    (↑(X.toTop (P := fun (X : TopCat.{u_1}) ↦
          TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
      Type u_1)

Tactic:
X.prop.2

Premises used: And.right, CompHausLike.prop
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
X Y : FintypeCat
f : X ⟶ Y
⊢ Continuous (X := X.obj) (Y := Y.obj)
    (⇑(@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _ f) :
      (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) X) →
        (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) Y)

Tactic:
fun_prop
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
⊢ toLightProfinite.Faithful

Tactic:
FintypeCat.toLightProfiniteFullyFaithful.faithful

Premises used: FintypeCat.toLightProfiniteFullyFaithful, CategoryTheory.Functor.FullyFaithful.faithful
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
⊢ toLightProfinite.Full

Tactic:
FintypeCat.toLightProfiniteFullyFaithful.full

Premises used: FintypeCat.toLightProfiniteFullyFaithful, CategoryTheory.Functor.FullyFaithful.full
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
X : FintypeCat
⊢ Finite.{u + 1}
    (↑((toLightProfinite.obj X).toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
      Type u)

Tactic:
inferInstanceAs (Finite X)

Premises used: _private.0.instFiniteCarrierToTopAndTotallyDisconnectedSpaceSecondCountableTopologyObjFintypeCatLightProfiniteToLightProfinite._proof_1
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
X : FintypeCat
⊢ Finite.{u + 1}
    (↑((LightProfinite.of X.obj).toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
      Type u)

Tactic:
inferInstanceAs (Finite X)

Premises used: _private.0.instFiniteCarrierToTopAndTotallyDisconnectedSpaceSecondCountableTopologyOfObj._proof_1
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
J : Type v
inst : SmallCategory J
F : J ⥤ LightProfinite
change :
  TotallyDisconnectedSpace
    (↑{u :
          (j : J) →
            (↑((F.obj j).toTop (P := fun (X : TopCat) ↦
                  TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
              Type (max u v)) |
          ∀ {i j : J} (f : i ⟶ j),
            (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
                    TopCat.carrier) (((F ⋙ lightProfiniteToCompHaus) ⋙ compHausToTop).map f) :
                  (a : (↑(((F ⋙ lightProfiniteToCompHaus) ⋙ compHausToTop).obj i) : Type (max u v))) →
                    (↑(((F ⋙ lightProfiniteToCompHaus) ⋙ compHausToTop).obj j) : Type (max u v)))
                (u i) =
              u j} :
      Type (max u v))
⊢ TotallyDisconnectedSpace
    (↑((CompHaus.limitCone.{v, u} (F ⋙ lightProfiniteToCompHaus)).pt.toTop (P := fun (x : TopCat) ↦ True)) :
      Type (max u v))

Tactic:
change TotallyDisconnectedSpace ({ u : ∀ j : J, F.obj j | _ } : Type _)

Premises used: setOf, Set.Elem, TotallyDisconnectedSpace
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
J : Type v
inst : SmallCategory J
F : J ⥤ LightProfinite
⊢ TotallyDisconnectedSpace
    (↑{u :
          (j : J) →
            (↑((F.obj j).toTop (P := fun (X : TopCat) ↦
                  TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
              Type (max u v)) |
          ∀ {i j : J} (f : i ⟶ j),
            (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
                    TopCat.carrier) (((F ⋙ lightProfiniteToCompHaus) ⋙ compHausToTop).map f) :
                  (a : (↑(((F ⋙ lightProfiniteToCompHaus) ⋙ compHausToTop).obj i) : Type (max u v))) →
                    (↑(((F ⋙ lightProfiniteToCompHaus) ⋙ compHausToTop).obj j) : Type (max u v)))
                (u i) =
              u j} :
      Type (max u v))

Tactic:
Subtype.totallyDisconnectedSpace

Premises used: Subtype.totallyDisconnectedSpace
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
J : Type v
inst : SmallCategory J
F : J ⥤ LightProfinite
⊢ TotallyDisconnectedSpace
    (↑{u :
          (j : J) →
            (↑((F.obj j).toTop (P := fun (X : TopCat) ↦
                  TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
              Type (max u v)) |
          ∀ {i j : J} (f : i ⟶ j),
            (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
                    TopCat.carrier) (((F ⋙ lightProfiniteToCompHaus) ⋙ compHausToTop).map f) :
                  (a : (↑(((F ⋙ lightProfiniteToCompHaus) ⋙ compHausToTop).obj i) : Type (max u v))) →
                    (↑(((F ⋙ lightProfiniteToCompHaus) ⋙ compHausToTop).obj j) : Type (max u v)))
                (u i) =
              u j} :
      Type (max u v))

Tactic:
exact Subtype.totallyDisconnectedSpace

Premises used: Subtype.totallyDisconnectedSpace
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
J : Type v
inst : SmallCategory J
inst_1 : CountableCategory J
F : J ⥤ LightProfinite
S : Cone F
x : S.pt ⟶ (limitCone.{v, u} F).pt
h : ∀ (j : J), x ≫ (limitCone.{v, u} F).π.app j = S.π.app j
a :
  lightProfiniteToCompHaus.map x =
    lightProfiniteToCompHaus.map
      (ConcreteCategory.ofHom (FC :=
        fun
          (x1 x2 :
            CompHausLike fun (X : TopCat) ↦
              TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v))) ↦
        C((↑(x1.toTop (P := fun (X : TopCat) ↦
                TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
            Type (max u v)),
          (↑(x2.toTop (P := fun (X : TopCat) ↦
                TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
            Type (max u v))))
        (CC :=
        fun
          (X :
            CompHausLike fun (X : TopCat) ↦
              TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v))) ↦
        (↑(X.toTop (P := fun (X : TopCat) ↦
              TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
          Type (max u v)))
        (TopCat.Hom.hom
          (InducedCategory.Hom.hom
            ((CompHaus.limitConeIsLimit (F ⋙ lightProfiniteToCompHaus)).lift (lightProfiniteToCompHaus.mapCone S)))))
⊢ x =
    ConcreteCategory.ofHom (FC :=
      fun
        (x1 x2 :
          CompHausLike fun (X : TopCat) ↦
            TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v))) ↦
      C((↑(x1.toTop (P := fun (X : TopCat) ↦
              TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
          Type (max u v)),
        (↑(x2.toTop (P := fun (X : TopCat) ↦
              TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
          Type (max u v))))
      (CC :=
      fun
        (X :
          CompHausLike fun (X : TopCat) ↦
            TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v))) ↦
      (↑(X.toTop (P := fun (X : TopCat) ↦
            TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
        Type (max u v)))
      (TopCat.Hom.hom
        (InducedCategory.Hom.hom
          ((CompHaus.limitConeIsLimit (F ⋙ lightProfiniteToCompHaus)).lift (lightProfiniteToCompHaus.mapCone S))))

Tactic:
apply lightProfiniteToCompHaus.map_injective

Premises used: lightProfiniteToCompHaus, CategoryTheory.Functor.map_injective
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
J : Type v
inst : SmallCategory J
inst_1 : CountableCategory J
F : J ⥤ LightProfinite
S : Cone F
x : S.pt ⟶ (limitCone.{v, u} F).pt
h : ∀ (j : J), x ≫ (limitCone.{v, u} F).π.app j = S.π.app j
a :
  ∀ (j : J),
    lightProfiniteToCompHaus.map x ≫ (CompHaus.limitCone.{v, u} (F ⋙ lightProfiniteToCompHaus)).π.app j =
      (lightProfiniteToCompHaus.mapCone S).π.app j
⊢ lightProfiniteToCompHaus.map x =
    lightProfiniteToCompHaus.map
      (ConcreteCategory.ofHom (FC :=
        fun
          (x1 x2 :
            CompHausLike fun (X : TopCat) ↦
              TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v))) ↦
        C((↑(x1.toTop (P := fun (X : TopCat) ↦
                TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
            Type (max u v)),
          (↑(x2.toTop (P := fun (X : TopCat) ↦
                TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
            Type (max u v))))
        (CC :=
        fun
          (X :
            CompHausLike fun (X : TopCat) ↦
              TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v))) ↦
        (↑(X.toTop (P := fun (X : TopCat) ↦
              TotallyDisconnectedSpace (↑X : Type (max u v)) ∧ SecondCountableTopology (↑X : Type (max u v)))) :
          Type (max u v)))
        (TopCat.Hom.hom
          (InducedCategory.Hom.hom
            ((CompHaus.limitConeIsLimit (F ⋙ lightProfiniteToCompHaus)).lift (lightProfiniteToCompHaus.mapCone S)))))

Tactic:
apply (CompHaus.limitConeIsLimit.{v, u} _).uniq (lightProfiniteToCompHaus.mapCone S)
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
J : Type v
inst : SmallCategory J
inst_1 : CountableCategory J
F : J ⥤ LightProfinite
S : Cone F
x : S.pt ⟶ (limitCone.{v, u} F).pt
h : ∀ (j : J), x ≫ (limitCone.{v, u} F).π.app j = S.π.app j
j : J
a :
  Eq (α := lightProfiniteToCompHaus.obj S.pt ⟶ lightProfiniteToCompHaus.obj (F.obj j))
    (ConcreteCategory.ofHom (FC := fun (x1 x2 : CompHausLike fun (x : TopCat) ↦ True) ↦
        C((↑(x1.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v)),
          (↑(x2.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v))))
        (CC := fun (X : CompHausLike fun (x : TopCat) ↦ True) ↦
        (↑(X.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v))) (TopCat.Hom.hom x.hom) ≫
      (CompHaus.limitCone.{v, u} (F ⋙ lightProfiniteToCompHaus)).π.app j)
    (ConcreteCategory.ofHom (FC := fun (x1 x2 : CompHausLike fun (x : TopCat) ↦ True) ↦
      C((↑(x1.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v)),
        (↑(x2.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v))))
      (CC := fun (X : CompHausLike fun (x : TopCat) ↦ True) ↦
      (↑(X.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v)))
      (TopCat.Hom.hom (x ≫ (limitCone.{v, u} F).π.app j).hom))
⊢ lightProfiniteToCompHaus.map x ≫ (CompHaus.limitCone.{v, u} (F ⋙ lightProfiniteToCompHaus)).π.app j =
    (lightProfiniteToCompHaus.mapCone S).π.app j

Tactic:
simp [← h]
-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
J : Type v
inst : SmallCategory J
inst_1 : CountableCategory J
F : J ⥤ LightProfinite
S : Cone F
x : S.pt ⟶ (limitCone.{v, u} F).pt
h : ∀ (j : J), x ≫ (limitCone.{v, u} F).π.app j = S.π.app j
j : J
⊢ Eq (α := lightProfiniteToCompHaus.obj S.pt ⟶ lightProfiniteToCompHaus.obj (F.obj j))
    (ConcreteCategory.ofHom (FC := fun (x1 x2 : CompHausLike fun (x : TopCat) ↦ True) ↦
        C((↑(x1.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v)),
          (↑(x2.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v))))
        (CC := fun (X : CompHausLike fun (x : TopCat) ↦ True) ↦
        (↑(X.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v))) (TopCat.Hom.hom x.hom) ≫
      (CompHaus.limitCone.{v, u} (F ⋙ lightProfiniteToCompHaus)).π.app j)
    (ConcreteCategory.ofHom (FC := fun (x1 x2 : CompHausLike fun (x : TopCat) ↦ True) ↦
      C((↑(x1.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v)),
        (↑(x2.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v))))
      (CC := fun (X : CompHausLike fun (x : TopCat) ↦ True) ↦
      (↑(X.toTop (P := fun (x : TopCat) ↦ True)) : Type (max u v)))
      (TopCat.Hom.hom (x ≫ (limitCone.{v, u} F).π.app j).hom))

Tactic:
rfl