-- Module: Mathlib.Algebra.Category.ModuleCat.Topology.Basic
Goal:
R : Type u
inst : Ring R
inst_1 : TopologicalSpace R
X : TopCat
h :
  Eq (α :=
    (↑(toModuleCat (Functor.obj (free R) X)) : Type (max u v)) →L[R]
      (↑(toModuleCat (Functor.obj (free R) X)) : Type (max u v)))
    (@hom _ (instCategory R)
        (fun (x1 x2 : TopModuleCat R) ↦ (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v)))
        (fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v)))
        (fun (X Y : TopModuleCat R) ↦
          (ContinuousLinearMap.funLike :
            FunLike ((↑X.toModuleCat : Type (max u v)) →L[R] (↑Y.toModuleCat : Type (max u v)))
              (↑X.toModuleCat : Type (max u v)) (↑Y.toModuleCat : Type (max u v))))
        _ _ _
        (CategoryStruct.comp (Y :=
            Functor.obj (free R)
              (Functor.obj
                (forget₂ (TopModuleCat R) (FC := fun (x1 x2 : TopModuleCat R) ↦
                  (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC :=
                  fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v))) TopCat (FD := fun (X Y : TopCat) ↦
                  C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CD := TopCat.carrier))
                (Functor.obj (free R) X)))
            (CategoryTheory.Functor.map (free R)
              (NatTrans.app
                {
                  app := fun (X : TopCat) ↦
                    TopCat.ofHom
                      { toFun := fun (x : (↑X : Type (max u v))) ↦ Finsupp.single x 1, continuous_toFun := ⋯ },
                  naturality := ⋯ }
                X))
            (NatTrans.app
              {
                app := fun (X : TopModuleCat R) ↦
                  ConcreteCategory.ofHom (FC := fun (x1 x2 : TopModuleCat R) ↦
                    (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC :=
                    fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v)))
                    {
                      toLinearMap :=
                        (Finsupp.lift (↑(Functor.obj (𝟭 (TopModuleCat R)) X).toModuleCat : Type (max u v)) R
                              (↑X.toModuleCat : Type (max u v)) :
                            (a :
                                (↑X.toModuleCat : Type (max u v)) →
                                  (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat : Type (max u v))) →
                              ((↑X.toModuleCat : Type (max u v)) →₀ R) →ₗ[R]
                                (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat : Type (max u v)))
                          id,
                      cont := ⋯ :
                      (↑(toModuleCat
                              (Functor.obj
                                (forget₂ (TopModuleCat R) (FC := fun (x1 x2 : TopModuleCat R) ↦
                                    (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC :=
                                    fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v))) TopCat (FD :=
                                    fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CD :=
                                    TopCat.carrier) ⋙
                                  free R)
                                X)) :
                          Type (max u v)) →L[R]
                        (↑(Functor.obj (𝟭 (TopModuleCat R)) X).toModuleCat : Type (max u v)) },
                naturality := ⋯ }
              (Functor.obj (free R) X)) :
          Quiver.Hom (V := TopModuleCat R) (Functor.obj (free R) X) (Functor.obj (free R) X)) :
      (↑(toModuleCat (Functor.obj (free R) X)) : Type (max u v)) →L[R]
        (↑(toModuleCat (Functor.obj (free R) X)) : Type (max u v)))
    (@hom _ (instCategory R)
        (fun (x1 x2 : TopModuleCat R) ↦ (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v)))
        (fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v)))
        (fun (X Y : TopModuleCat R) ↦
          (ContinuousLinearMap.funLike :
            FunLike ((↑X.toModuleCat : Type (max u v)) →L[R] (↑Y.toModuleCat : Type (max u v)))
              (↑X.toModuleCat : Type (max u v)) (↑Y.toModuleCat : Type (max u v))))
        _ _ _
        (𝟙 (Functor.obj (free R) X) :
          Quiver.Hom (V := TopModuleCat R) (Functor.obj (free R) X) (Functor.obj (free R) X)) :
      (↑(toModuleCat (Functor.obj (free R) X)) : Type (max u v)) →L[R]
        (↑(toModuleCat (Functor.obj (free R) X)) : Type (max u v)))
⊢ Eq (α := Quiver.Hom (V := TopModuleCat R) (Functor.obj (free R) X) (Functor.obj (free R) X))
    (CategoryStruct.comp (Y :=
        Functor.obj (free R)
          (Functor.obj
            (forget₂ (TopModuleCat R) (FC := fun (x1 x2 : TopModuleCat R) ↦
              (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC :=
              fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v))) TopCat (FD := fun (X Y : TopCat) ↦
              C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CD := TopCat.carrier))
            (Functor.obj (free R) X)))
        (CategoryTheory.Functor.map (free R)
          (NatTrans.app
            {
              app := fun (X : TopCat) ↦
                TopCat.ofHom { toFun := fun (x : (↑X : Type (max u v))) ↦ Finsupp.single x 1, continuous_toFun := ⋯ },
              naturality := ⋯ }
            X))
        (NatTrans.app
          {
            app := fun (X : TopModuleCat R) ↦
              ConcreteCategory.ofHom (FC := fun (x1 x2 : TopModuleCat R) ↦
                (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC :=
                fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v)))
                {
                  toLinearMap :=
                    (Finsupp.lift (↑(Functor.obj (𝟭 (TopModuleCat R)) X).toModuleCat : Type (max u v)) R
                          (↑X.toModuleCat : Type (max u v)) :
                        (a :
                            (↑X.toModuleCat : Type (max u v)) →
                              (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat : Type (max u v))) →
                          ((↑X.toModuleCat : Type (max u v)) →₀ R) →ₗ[R]
                            (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat : Type (max u v)))
                      id,
                  cont := ⋯ :
                  (↑(toModuleCat
                          (Functor.obj
                            (forget₂ (TopModuleCat R) (FC := fun (x1 x2 : TopModuleCat R) ↦
                                (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC :=
                                fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v))) TopCat (FD :=
                                fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CD :=
                                TopCat.carrier) ⋙
                              free R)
                            X)) :
                      Type (max u v)) →L[R]
                    (↑(Functor.obj (𝟭 (TopModuleCat R)) X).toModuleCat : Type (max u v)) },
            naturality := ⋯ }
          (Functor.obj (free R) X)) :
      Quiver.Hom (V := TopModuleCat R) (Functor.obj (free R) X) (Functor.obj (free R) X))
    (𝟙 (Functor.obj (free R) X) : Quiver.Hom (V := TopModuleCat R) (Functor.obj (free R) X) (Functor.obj (free R) X))

Tactic:
ext1