-- Lean 4 proof step from Mathlib.Topology.Category.UniformSpace (goal)
Goal:
X' X : UniformSpaceCat
Y : CpltSepUniformSpace
f : X' ⟶ X
g :
  X ⟶
    Functor.obj
      (forget₂ CpltSepUniformSpace (FC := fun (x1 x2 : CpltSepUniformSpace.{u_1}) ↦
        { f : x1.α → x2.α // UniformContinuous f }) (CC := α) UniformSpaceCat (FD :=
        fun (x1 x2 : UniformSpaceCat.{u_1}) ↦ { f : x1.carrier → x2.carrier // UniformContinuous f }) (CD := carrier))
      Y
⊢ Eq (α := completionFunctor.obj X' ⟶ Y)
    ((Equiv.symm
            { toFun := fun (f : completionFunctor.obj X' ⟶ Y) ↦ X'.completionHom ≫ f.hom,
              invFun :=
                fun
                  (f :
                    X' ⟶
                      Functor.obj
                        (forget₂ CpltSepUniformSpace (FC := fun (x1 x2 : CpltSepUniformSpace.{u_1}) ↦
                          { f : x1.α → x2.α // UniformContinuous f }) (CC := α) UniformSpaceCat (FD :=
                          fun (x1 x2 : UniformSpaceCat.{u_1}) ↦ { f : x1.carrier → x2.carrier // UniformContinuous f })
                          (CD := carrier))
                        Y) ↦
                extensionHom f,
              left_inv := ⋯, right_inv := ⋯ } :
          (a : X' ⟶ (forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y) → completionFunctor.obj X' ⟶ Y)
        (f ≫ g) :
      completionFunctor.obj X' ⟶ Y)
    (completionFunctor.map f ≫
        (Equiv.symm
              { toFun := fun (f : completionFunctor.obj X ⟶ Y) ↦ X.completionHom ≫ f.hom,
                invFun :=
                  fun
                    (f :
                      X ⟶
                        Functor.obj
                          (forget₂ CpltSepUniformSpace (FC := fun (x1 x2 : CpltSepUniformSpace.{u_1}) ↦
                            { f : x1.α → x2.α // UniformContinuous f }) (CC := α) UniformSpaceCat (FD :=
                            fun (x1 x2 : UniformSpaceCat.{u_1}) ↦
                            { f : x1.carrier → x2.carrier // UniformContinuous f }) (CD := carrier))
                          Y) ↦
                  extensionHom f,
                left_inv := ⋯, right_inv := ⋯ } :
            (a : X ⟶ (forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y) → completionFunctor.obj X ⟶ Y)
          g :
      completionFunctor.obj X' ⟶ Y)
have h : ∀ (x : (completionFunctor.obj X').α),
  Eq (α := Y.α)
    ((↑(@ConcreteCategory.hom _ category
              (fun (x1 x2 : CpltSepUniformSpace.{u_1}) ↦ { f : x1.α → x2.α // UniformContinuous f }) α _ _
              (completionFunctor.obj X') Y
              ((Equiv.symm
                    { toFun := fun (f : completionFunctor.obj X' ⟶ Y) ↦ X'.completionHom ≫ f.hom,
                      invFun :=
                        fun
                          (f :
                            X' ⟶
                              Functor.obj
                                (forget₂ CpltSepUniformSpace (FC := fun (x1 x2 : CpltSepUniformSpace.{u_1}) ↦
                                  { f : x1.α → x2.α // UniformContinuous f }) (CC := α) UniformSpaceCat (FD :=
                                  fun (x1 x2 : UniformSpaceCat.{u_1}) ↦
                                  { f : x1.carrier → x2.carrier // UniformContinuous f }) (CD := carrier))
                                Y) ↦
                        extensionHom f,
                      left_inv := sorry, right_inv := sorry :
                      (completionFunctor.obj X' ⟶ Y) ≃
                        (X' ⟶
                          Functor.obj
                            (forget₂ CpltSepUniformSpace (FC := fun (x1 x2 : CpltSepUniformSpace.{u_1}) ↦
                              { f : x1.α → x2.α // UniformContinuous f }) (CC := α) UniformSpaceCat (FD :=
                              fun (x1 x2 : UniformSpaceCat.{u_1}) ↦
                              { f : x1.carrier → x2.carrier // UniformContinuous f }) (CD := carrier))
                            Y) } :
                  (a : X' ⟶ (forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y) → completionFunctor.obj X' ⟶ Y)
                (f ≫ g))) :
          (completionFunctor.obj X').α → Y.α)
        x :
      Y.α)
    ((↑(ConcreteCategory.hom (FC := fun (x1 x2 : CpltSepUniformSpace.{u_1}) ↦
              { f : x1.α → x2.α // UniformContinuous f }) (CC := α) (Y := Y)
              (completionFunctor.map f ≫
                (Equiv.symm
                      { toFun := fun (f : completionFunctor.obj X ⟶ Y) ↦ X.completionHom ≫ f.hom,
                        invFun :=
                          fun
                            (f :
                              X ⟶
                                Functor.obj
                                  (forget₂ CpltSepUniformSpace (FC := fun (x1 x2 : CpltSepUniformSpace.{u_1}) ↦
                                    { f : x1.α → x2.α // UniformContinuous f }) (CC := α) UniformSpaceCat (FD :=
                                    fun (x1 x2 : UniformSpaceCat.{u_1}) ↦
                                    { f : x1.carrier → x2.carrier // UniformContinuous f }) (CD := carrier))
                                  Y) ↦
                          extensionHom f,
                        left_inv := sorry, right_inv := sorry :
                        (completionFunctor.obj X ⟶ Y) ≃
                          (X ⟶
                            Functor.obj
                              (forget₂ CpltSepUniformSpace (FC := fun (x1 x2 : CpltSepUniformSpace.{u_1}) ↦
                                { f : x1.α → x2.α // UniformContinuous f }) (CC := α) UniformSpaceCat (FD :=
                                fun (x1 x2 : UniformSpaceCat.{u_1}) ↦
                                { f : x1.carrier → x2.carrier // UniformContinuous f }) (CD := carrier))
                              Y) } :
                    (a : X ⟶ (forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y) → completionFunctor.obj X ⟶ Y)
                  g)) :
          (completionFunctor.obj X').α → Y.α)
        x :
      Y.α)
Tactic: ext x