-- Module: Mathlib.Topology.Category.UniformSpace
Goal:
X : UniformSpaceCat
Y : CpltSepUniformSpace
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
x : X.carrier
h :
  ∀
    (val :
      X.carrier →
        (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).carrier)
    (property : UniformContinuous val),
    Eq (α :=
      (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).carrier)
      ((@ConcreteCategory.hom _ instLargeCategory
              (fun (x1 x2 : UniformSpaceCat.{u_1}) ↦ { f : x1.carrier → x2.carrier // UniformContinuous f }) carrier _ _
              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)
              ((fun (f : completionFunctor.obj X ⟶ Y) ↦ X.completionHom ≫ f.hom)
                ((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)
                  {
                    hom' :=
                      Subtype.mk (p :=
                        fun
                          (f :
                            X.carrier →
                              (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).carrier) ↦
                        UniformContinuous f) val property })) :
            (a : X.carrier) → ((forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y).carrier)
          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).carrier)
      ((@ConcreteCategory.hom _ instLargeCategory
              (fun (x1 x2 : UniformSpaceCat.{u_1}) ↦ { f : x1.carrier → x2.carrier // UniformContinuous f }) carrier _ _
              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)
              {
                hom' :=
                  Subtype.mk (p :=
                    fun
                      (f :
                        X.carrier →
                          (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).carrier) ↦
                    UniformContinuous f) val property :
                X.Hom
                  (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.carrier) → ((forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y).carrier)
          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).carrier)
⊢ Eq (α :=
    (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).carrier)
    ((@ConcreteCategory.hom _ instLargeCategory
            (fun (x1 x2 : UniformSpaceCat.{u_1}) ↦ { f : x1.carrier → x2.carrier // UniformContinuous f }) carrier _ _ 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)
            ((fun (f : completionFunctor.obj X ⟶ Y) ↦ X.completionHom ≫ f.hom)
              ((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)
                f)) :
          (a : X.carrier) → ((forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y).carrier)
        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).carrier)
    ((ConcreteCategory.hom (FC := fun (x1 x2 : UniformSpaceCat.{u_1}) ↦
            { f : x1.carrier → x2.carrier // UniformContinuous f }) (CC := carrier) f :
          (a : X.carrier) → ((forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y).carrier)
        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).carrier)

Tactic:
rcases f with ⟨⟨_, _⟩⟩