-- Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion
Goal:
G : GrpCat
P : ProfiniteGrp.{u}
f :
  G ⟶
    GrpCat.of (↑(P.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)
X Y :
  OpenNormalSubgroup
    (↑(P.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)
g : X ⟶ Y
x :
  (j : FiniteIndexNormalSubgroup (↑G : Type u)) →
    (↑((((diagram G ⋙
                  forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                    (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                        Type u) →ₜ*
                      (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                        Type u))
                    (CC := fun (X : ProfiniteGrp.{u}) ↦
                    (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                      Type u))
                    Profinite (FD :=
                    fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                    C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
                      (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
                    (CD := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                    (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))) ⋙
                profiniteToCompHaus) ⋙
              compHausToTop).obj
          j) :
      Type u)
hx :
  Membership.mem (γ :=
    Set
      ((j : FiniteIndexNormalSubgroup (↑G : Type u)) →
        (↑((((diagram G ⋙
                      forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                        (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                            Type u) →ₜ*
                          (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                            Type u))
                        (CC := fun (X : ProfiniteGrp.{u}) ↦
                        (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                              TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                          Type u))
                        Profinite (FD :=
                        fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                        C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                            Type u),
                          (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                            Type u)))
                        (CD :=
                        fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                        (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                          Type u))) ⋙
                    profiniteToCompHaus) ⋙
                  compHausToTop).obj
              j) :
          Type u)))
    {u :
      (j : FiniteIndexNormalSubgroup (↑G : Type u)) →
        (↑((((diagram G ⋙
                      forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                        (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                            Type u) →ₜ*
                          (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                            Type u))
                        (CC := fun (X : ProfiniteGrp.{u}) ↦
                        (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                              TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                          Type u))
                        Profinite (FD :=
                        fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                        C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                            Type u),
                          (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                            Type u)))
                        (CD :=
                        fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                        (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                          Type u))) ⋙
                    profiniteToCompHaus) ⋙
                  compHausToTop).obj
              j) :
          Type u) |
      ∀ {i j : FiniteIndexNormalSubgroup (↑G : Type u)} (f : i ⟶ j),
        (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                ((((diagram G ⋙
                          forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                            (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                    TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                Type u) →ₜ*
                              (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                    TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                Type u))
                            (CC := fun (X : ProfiniteGrp.{u}) ↦
                            (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                  TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                              Type u))
                            Profinite (FD :=
                            fun
                              (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                            C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                Type u),
                              (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                Type u)))
                            (CD :=
                            fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                            (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                              Type u))) ⋙
                        profiniteToCompHaus) ⋙
                      compHausToTop).map
                  f) :
              (a :
                  (↑((((diagram G ⋙ forget₂ ProfiniteGrp.{u} Profinite) ⋙ profiniteToCompHaus) ⋙ compHausToTop).obj i) :
                    Type u)) →
                (↑((((diagram G ⋙ forget₂ ProfiniteGrp.{u} Profinite) ⋙ profiniteToCompHaus) ⋙ compHausToTop).obj j) :
                  Type u))
            (u i) =
          u j}
    x
hf :
  Eq (α :=
    (fun (X : FiniteGrp.{u}) ↦ (↑X.toGrp : Type u))
      (FiniteGrp.of
        ((↑(P.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u) ⧸
          (↑Y.toOpenSubgroup : Subgroup (↑P.toProfinite.toTop : Type u)))))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : FiniteGrp.{u}) ↦ (↑x1.toGrp : Type u) →* (↑x2.toGrp : Type u)) (CC :=
            fun (X : FiniteGrp.{u}) ↦ (↑X.toGrp : Type u)) (quotientMap f Y) :
          (a :
              (fun (X : FiniteGrp.{u}) ↦ (↑X.toGrp : Type u))
                (FiniteGrp.of ((↑G : Type u) ⧸ (preimage f Y).toSubgroup))) →
            (fun (X : FiniteGrp.{u}) ↦ (↑X.toGrp : Type u))
              (FiniteGrp.of
                ((↑P.toProfinite.toTop : Type u) ⧸ (↑Y.toOpenSubgroup : Subgroup (↑P.toProfinite.toTop : Type u)))))
        (x (preimage f Y)) :
      (fun (X : FiniteGrp.{u}) ↦ (↑X.toGrp : Type u))
        (FiniteGrp.of
          ((↑(P.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u) ⧸
            (↑Y.toOpenSubgroup : Subgroup (↑P.toProfinite.toTop : Type u)))))
    ((Hom.hom (P.diagram.map g) :
          (a : (↑(P.diagram.obj X).toProfinite.toTop : Type u)) → (↑(P.diagram.obj Y).toProfinite.toTop : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : FiniteGrp.{u}) ↦ (↑x1.toGrp : Type u) →* (↑x2.toGrp : Type u)) (CC :=
              fun (X : FiniteGrp.{u}) ↦ (↑X.toGrp : Type u)) (quotientMap f X) :
            (a :
                (fun (X : FiniteGrp.{u}) ↦ (↑X.toGrp : Type u))
                  (FiniteGrp.of ((↑G : Type u) ⧸ (preimage f X).toSubgroup))) →
              (fun (X : FiniteGrp.{u}) ↦ (↑X.toGrp : Type u))
                (FiniteGrp.of
                  ((↑P.toProfinite.toTop : Type u) ⧸ (↑X.toOpenSubgroup : Subgroup (↑P.toProfinite.toTop : Type u)))))
          (x (preimage f X))) :
      (↑((P.diagram.obj Y).toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
        Type u))
⊢ Eq (α :=
    (↑((P.diagram.obj Y).toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
      Type u))
    ((Hom.hom
            (CategoryTheory.Functor.map
                (Functor.obj
                  (Functor.const
                    (OpenNormalSubgroup
                      (↑(P.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                        Type u)))
                  (limitCone (diagram G)).pt)
                g ≫
              (limitCone (diagram G)).π.app (preimage f Y) ≫ ofFiniteGrpHom (quotientMap f Y)) :
          (a :
              (↑(((Functor.const (OpenNormalSubgroup (↑P.toProfinite.toTop : Type u))).obj
                            (limitCone (diagram G)).pt).obj
                        X).toProfinite.toTop :
                Type u)) →
            (↑(P.diagram.obj Y).toProfinite.toTop : Type u))
        (Subtype.mk (p :=
          fun
            (x :
              (j : FiniteIndexNormalSubgroup (↑G : Type u)) →
                (↑((((diagram G ⋙
                              forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u) →ₜ*
                                  (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u))
                                (CC := fun (X : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                      TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))
                                Profinite (FD :=
                                fun
                                  (x1 x2 :
                                    CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u),
                                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u)))
                                (CD :=
                                fun
                                  (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))) ⋙
                            profiniteToCompHaus) ⋙
                          compHausToTop).obj
                      j) :
                  Type u)) ↦
          Membership.mem (γ :=
            Set
              ((j : FiniteIndexNormalSubgroup (↑G : Type u)) →
                (↑((((diagram G ⋙
                              forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u) →ₜ*
                                  (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u))
                                (CC := fun (X : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                      TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))
                                Profinite (FD :=
                                fun
                                  (x1 x2 :
                                    CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u),
                                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u)))
                                (CD :=
                                fun
                                  (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))) ⋙
                            profiniteToCompHaus) ⋙
                          compHausToTop).obj
                      j) :
                  Type u)))
            {u :
              (j : FiniteIndexNormalSubgroup (↑G : Type u)) →
                (↑((((diagram G ⋙
                              forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u) →ₜ*
                                  (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u))
                                (CC := fun (X : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                      TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))
                                Profinite (FD :=
                                fun
                                  (x1 x2 :
                                    CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u),
                                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u)))
                                (CD :=
                                fun
                                  (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))) ⋙
                            profiniteToCompHaus) ⋙
                          compHausToTop).obj
                      j) :
                  Type u) |
              ∀ {i j : FiniteIndexNormalSubgroup (↑G : Type u)} (f : i ⟶ j),
                (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                        TopCat.carrier)
                        ((((diagram G ⋙
                                  forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                                    (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                            TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                        Type u) →ₜ*
                                      (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                            TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                        Type u))
                                    (CC := fun (X : ProfiniteGrp.{u}) ↦
                                    (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                          TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                      Type u))
                                    Profinite (FD :=
                                    fun
                                      (x1 x2 :
                                        CompHausLike fun (X : TopCat.{u}) ↦
                                          TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                    C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                                            TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                        Type u),
                                      (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                                            TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                        Type u)))
                                    (CD :=
                                    fun
                                      (X :
                                        CompHausLike fun (X : TopCat.{u}) ↦
                                          TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                    (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                                          TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                      Type u))) ⋙
                                profiniteToCompHaus) ⋙
                              compHausToTop).map
                          f) :
                      (a :
                          (↑((((diagram G ⋙ forget₂ ProfiniteGrp.{u} Profinite) ⋙ profiniteToCompHaus) ⋙
                                    compHausToTop).obj
                                i) :
                            Type u)) →
                        (↑((((diagram G ⋙ forget₂ ProfiniteGrp.{u} Profinite) ⋙ profiniteToCompHaus) ⋙
                                  compHausToTop).obj
                              j) :
                          Type u))
                    (u i) =
                  u j}
            x)
          x hx) :
      (↑((P.diagram.obj Y).toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
        Type u))
    ((Hom.hom (((limitCone (diagram G)).π.app (preimage f X) ≫ ofFiniteGrpHom (quotientMap f X)) ≫ P.diagram.map g) :
          (a :
              (↑(((Functor.const (OpenNormalSubgroup (↑P.toProfinite.toTop : Type u))).obj
                            (limitCone (diagram G)).pt).obj
                        X).toProfinite.toTop :
                Type u)) →
            (↑(P.diagram.obj Y).toProfinite.toTop : Type u))
        (Subtype.mk (p :=
          fun
            (x :
              (j : FiniteIndexNormalSubgroup (↑G : Type u)) →
                (↑((((diagram G ⋙
                              forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u) →ₜ*
                                  (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u))
                                (CC := fun (X : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                      TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))
                                Profinite (FD :=
                                fun
                                  (x1 x2 :
                                    CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u),
                                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u)))
                                (CD :=
                                fun
                                  (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))) ⋙
                            profiniteToCompHaus) ⋙
                          compHausToTop).obj
                      j) :
                  Type u)) ↦
          Membership.mem (γ :=
            Set
              ((j : FiniteIndexNormalSubgroup (↑G : Type u)) →
                (↑((((diagram G ⋙
                              forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u) →ₜ*
                                  (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u))
                                (CC := fun (X : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                      TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))
                                Profinite (FD :=
                                fun
                                  (x1 x2 :
                                    CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u),
                                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u)))
                                (CD :=
                                fun
                                  (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))) ⋙
                            profiniteToCompHaus) ⋙
                          compHausToTop).obj
                      j) :
                  Type u)))
            {u :
              (j : FiniteIndexNormalSubgroup (↑G : Type u)) →
                (↑((((diagram G ⋙
                              forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u) →ₜ*
                                  (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u))
                                (CC := fun (X : ProfiniteGrp.{u}) ↦
                                (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                      TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))
                                Profinite (FD :=
                                fun
                                  (x1 x2 :
                                    CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u),
                                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                                        TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                    Type u)))
                                (CD :=
                                fun
                                  (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                  Type u))) ⋙
                            profiniteToCompHaus) ⋙
                          compHausToTop).obj
                      j) :
                  Type u) |
              ∀ {i j : FiniteIndexNormalSubgroup (↑G : Type u)} (f : i ⟶ j),
                (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                        TopCat.carrier)
                        ((((diagram G ⋙
                                  forget₂ ProfiniteGrp.{u} (FC := fun (X Y : ProfiniteGrp.{u}) ↦
                                    (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                            TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                        Type u) →ₜ*
                                      (↑(Y.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                            TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                        Type u))
                                    (CC := fun (X : ProfiniteGrp.{u}) ↦
                                    (↑(X.toProfinite.toTop (P := fun (X : TopCat.{u}) ↦
                                          TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                      Type u))
                                    Profinite (FD :=
                                    fun
                                      (x1 x2 :
                                        CompHausLike fun (X : TopCat.{u}) ↦
                                          TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                    C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                                            TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                        Type u),
                                      (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                                            TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                        Type u)))
                                    (CD :=
                                    fun
                                      (X :
                                        CompHausLike fun (X : TopCat.{u}) ↦
                                          TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                                    (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                                          TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                                      Type u))) ⋙
                                profiniteToCompHaus) ⋙
                              compHausToTop).map
                          f) :
                      (a :
                          (↑((((diagram G ⋙ forget₂ ProfiniteGrp.{u} Profinite) ⋙ profiniteToCompHaus) ⋙
                                    compHausToTop).obj
                                i) :
                            Type u)) →
                        (↑((((diagram G ⋙ forget₂ ProfiniteGrp.{u} Profinite) ⋙ profiniteToCompHaus) ⋙
                                  compHausToTop).obj
                              j) :
                          Type u))
                    (u i) =
                  u j}
            x)
          x hx) :
      (↑((P.diagram.obj Y).toProfinite.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
        Type u))

Tactic:
change quotientMap f Y (x <| preimage f Y) =
        P.diagram.map g (quotientMap _ _ <| x <| preimage f X)

Premises used: CategoryTheory.ConcreteCategory.hom, CategoryTheory.Functor.map, ProfiniteGrp.ProfiniteCompletion.quotientMap, ProfiniteGrp.diagram, ProfiniteGrp.Hom.hom, DFunLike.coe, ProfiniteGrp.ProfiniteCompletion.preimage