-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Topology.Basic (goal)
Goal:
R : Type u
inst : Ring R
inst_1 : TopologicalSpace R
X : TopCat
⊢ 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))
have 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 := sorry },
                naturality := sorry }
              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 := sorry :
                    (↑(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 := sorry }
            (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)))
Tactic: ext1