/-- For any `S`-module Y, there is a natural `R`-linear map from `S ⨂ Y` to `Y` by
`s ⊗ y ↦ s • y`
-/
@[simps apply]
def Counit.map {Y} : (restrictScalars f ⋙ extendScalars f).obj Y ⟶ Y := by
  letI m1 : Module R S := Module.compHom S f
  letI m2 : Module R Y := Module.compHom Y f
  refine'
    {toFun := TensorProduct.lift
      {toFun := fun s : S => {toFun := fun y : Y => s • y, map_add' := smul_add _, map_smul' := _},
        map_add' := _, map_smul' := _}, map_add' := _, map_smul' := _}
  · intros r y
    dsimp
    change s • f r • y = f r • s • y
    rw [← mul_smul, mul_comm, mul_smul]
  · intros s₁ s₂
    ext y
    change (s₁ + s₂) • y = s₁ • y + s₂ • y
    rw [add_smul]
  · intros r s
    ext y
    change (f r • s) • y = (f r) • s • y
    rw [smul_eq_mul,mul_smul]
  · intros
    rw [map_add]
  · intro s z
    

Proof state:
case refine'_5
R✝ : Type u₁
S✝ : Type u₂
inst✝³ : CommRing R✝
inst✝² : CommRing S✝
f✝ : R✝ →+* S✝
R : Type u₁
S : Type u₂
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
Y : ModuleCat S
m1 : Module R S := Module.compHom S f
m2 : Module R ↑Y := Module.compHom (↑Y) f
s : S
z : ↑((restrictScalars f ⋙ extendScalars f).obj Y)
⊢ AddHom.toFun
      {
        toFun :=
          ⇑(lift
              {
                toAddHom :=
                  {
                    toFun := fun s =>
                      {
                        toAddHom :=
                          { toFun := fun y => s • y,
                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                        map_smul' :=
                          (_ :
                            ∀ (r : R) (y : ↑Y),
                              AddHom.toFun
                                  { toFun := fun y => s • y,
                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                  (r • y) =
                                (RingHom.id R) r •
                                  AddHom.toFun
                                    { toFun := fun y => s • y,
                                      map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                    y) },
                    map_add' :=
                      (_ :
                        ∀ (s₁ s₂ : S),
                          (fun s =>
                                {
                                  toAddHom :=
                                    { toFun := fun y => s • y,
                                      map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                  map_smul' :=
                                    (_ :
                                      ∀ (r : R) (y : ↑Y),
                                        AddHom.toFun
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                            (r • y) =
                                          (RingHom.id R) r •
                                            AddHom.toFun
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                              y) })
                              (s₁ + s₂) =
                            (fun s =>
                                  {
                                    toAddHom :=
                                      { toFun := fun y => s • y,
                                        map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                    map_smul' :=
                                      (_ :
                                        ∀ (r : R) (y : ↑Y),
                                          AddHom.toFun
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                              (r • y) =
                                            (RingHom.id R) r •
                                              AddHom.toFun
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                y) })
                                s₁ +
                              (fun s =>
                                  {
                                    toAddHom :=
                                      { toFun := fun y => s • y,
                                        map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                    map_smul' :=
                                      (_ :
                                        ∀ (r : R) (y : ↑Y),
                                          AddHom.toFun
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                              (r • y) =
                                            (RingHom.id R) r •
                                              AddHom.toFun
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                y) })
                                s₂) },
                map_smul' :=
                  (_ :
                    ∀ (r : R) (s : S),
                      AddHom.toFun
                          {
                            toFun := fun s =>
                              {
                                toAddHom :=
                                  { toFun := fun y => s • y,
                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                map_smul' :=
                                  (_ :
                                    ∀ (r : R) (y : ↑Y),
                                      AddHom.toFun
                                          { toFun := fun y => s • y,
                                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                          (r • y) =
                                        (RingHom.id R) r •
                                          AddHom.toFun
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                            y) },
                            map_add' :=
                              (_ :
                                ∀ (s₁ s₂ : S),
                                  (fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) })
                                      (s₁ + s₂) =
                                    (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        s₁ +
                                      (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        s₂) }
                          (r • s) =
                        (RingHom.id R) r •
                          AddHom.toFun
                            {
                              toFun := fun s =>
                                {
                                  toAddHom :=
                                    { toFun := fun y => s • y,
                                      map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                  map_smul' :=
                                    (_ :
                                      ∀ (r : R) (y : ↑Y),
                                        AddHom.toFun
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                            (r • y) =
                                          (RingHom.id R) r •
                                            AddHom.toFun
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                              y) },
                              map_add' :=
                                (_ :
                                  ∀ (s₁ s₂ : S),
                                    (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        (s₁ + s₂) =
                                      (fun s =>
                                            {
                                              toAddHom :=
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (y : ↑Y),
                                                    AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        (r • y) =
                                                      (RingHom.id R) r •
                                                        AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          y) })
                                          s₁ +
                                        (fun s =>
                                            {
                                              toAddHom :=
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (y : ↑Y),
                                                    AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        (r • y) =
                                                      (RingHom.id R) r •
                                                        AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          y) })
                                          s₂) }
                            s) }),
        map_add' :=
          (_ :
            ∀ (x y : ↑((restrictScalars f ⋙ extendScalars f).obj Y)),
              (lift
                    {
                      toAddHom :=
                        {
                          toFun := fun s =>
                            {
                              toAddHom :=
                                { toFun := fun y => s • y,
                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                              map_smul' :=
                                (_ :
                                  ∀ (r : R) (y : ↑Y),
                                    AddHom.toFun
                                        { toFun := fun y => s • y,
                                          map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                        (r • y) =
                                      (RingHom.id R) r •
                                        AddHom.toFun
                                          { toFun := fun y => s • y,
                                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                          y) },
                          map_add' :=
                            (_ :
                              ∀ (s₁ s₂ : S),
                                (fun s =>
                                      {
                                        toAddHom :=
                                          { toFun := fun y => s • y,
                                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                        map_smul' :=
                                          (_ :
                                            ∀ (r : R) (y : ↑Y),
                                              AddHom.toFun
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                  (r • y) =
                                                (RingHom.id R) r •
                                                  AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    y) })
                                    (s₁ + s₂) =
                                  (fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) })
                                      s₁ +
                                    (fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) })
                                      s₂) },
                      map_smul' :=
                        (_ :
                          ∀ (r : R) (s : S),
                            AddHom.toFun
                                {
                                  toFun := fun s =>
                                    {
                                      toAddHom :=
                                        { toFun := fun y => s • y,
                                          map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                      map_smul' :=
                                        (_ :
                                          ∀ (r : R) (y : ↑Y),
                                            AddHom.toFun
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                (r • y) =
                                              (RingHom.id R) r •
                                                AddHom.toFun
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                  y) },
                                  map_add' :=
                                    (_ :
                                      ∀ (s₁ s₂ : S),
                                        (fun s =>
                                              {
                                                toAddHom :=
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                map_smul' :=
                                                  (_ :
                                                    ∀ (r : R) (y : ↑Y),
                                                      AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          (r • y) =
                                                        (RingHom.id R) r •
                                                          AddHom.toFun
                                                            { toFun := fun y => s • y,
                                                              map_add' :=
                                                                (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                            y) })
                                            (s₁ + s₂) =
                                          (fun s =>
                                                {
                                                  toAddHom :=
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                  map_smul' :=
                                                    (_ :
                                                      ∀ (r : R) (y : ↑Y),
                                                        AddHom.toFun
                                                            { toFun := fun y => s • y,
                                                              map_add' :=
                                                                (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                            (r • y) =
                                                          (RingHom.id R) r •
                                                            AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              y) })
                                              s₁ +
                                            (fun s =>
                                                {
                                                  toAddHom :=
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                  map_smul' :=
                                                    (_ :
                                                      ∀ (r : R) (y : ↑Y),
                                                        AddHom.toFun
                                                            { toFun := fun y => s • y,
                                                              map_add' :=
                                                                (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                            (r • y) =
                                                          (RingHom.id R) r •
                                                            AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              y) })
                                              s₂) }
                                (r • s) =
                              (RingHom.id R) r •
                                AddHom.toFun
                                  {
                                    toFun := fun s =>
                                      {
                                        toAddHom :=
                                          { toFun := fun y => s • y,
                                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                        map_smul' :=
                                          (_ :
                                            ∀ (r : R) (y : ↑Y),
                                              AddHom.toFun
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                  (r • y) =
                                                (RingHom.id R) r •
                                                  AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    y) },
                                    map_add' :=
                                      (_ :
                                        ∀ (s₁ s₂ : S),
                                          (fun s =>
                                                {
                                                  toAddHom :=
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                  map_smul' :=
                                                    (_ :
                                                      ∀ (r : R) (y : ↑Y),
                                                        AddHom.toFun
                                                            { toFun := fun y => s • y,
                                                              map_add' :=
                                                                (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                            (r • y) =
                                                          (RingHom.id R) r •
                                                            AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              y) })
                                              (s₁ + s₂) =
                                            (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                s₁ +
                                              (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                s₂) }
                                  s) })
                  (x + y) =
                (lift
                      {
                        toAddHom :=
                          {
                            toFun := fun s =>
                              {
                                toAddHom :=
                                  { toFun := fun y => s • y,
                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                map_smul' :=
                                  (_ :
                                    ∀ (r : R) (y : ↑Y),
                                      AddHom.toFun
                                          { toFun := fun y => s • y,
                                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                          (r • y) =
                                        (RingHom.id R) r •
                                          AddHom.toFun
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                            y) },
                            map_add' :=
                              (_ :
                                ∀ (s₁ s₂ : S),
                                  (fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) })
                                      (s₁ + s₂) =
                                    (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        s₁ +
                                      (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        s₂) },
                        map_smul' :=
                          (_ :
                            ∀ (r : R) (s : S),
                              AddHom.toFun
                                  {
                                    toFun := fun s =>
                                      {
                                        toAddHom :=
                                          { toFun := fun y => s • y,
                                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                        map_smul' :=
                                          (_ :
                                            ∀ (r : R) (y : ↑Y),
                                              AddHom.toFun
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                  (r • y) =
                                                (RingHom.id R) r •
                                                  AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    y) },
                                    map_add' :=
                                      (_ :
                                        ∀ (s₁ s₂ : S),
                                          (fun s =>
                                                {
                                                  toAddHom :=
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                  map_smul' :=
                                                    (_ :
                                                      ∀ (r : R) (y : ↑Y),
                                                        AddHom.toFun
                                                            { toFun := fun y => s • y,
                                                              map_add' :=
                                                                (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                            (r • y) =
                                                          (RingHom.id R) r •
                                                            AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              y) })
                                              (s₁ + s₂) =
                                            (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                s₁ +
                                              (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                s₂) }
                                  (r • s) =
                                (RingHom.id R) r •
                                  AddHom.toFun
                                    {
                                      toFun := fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) },
                                      map_add' :=
                                        (_ :
                                          ∀ (s₁ s₂ : S),
                                            (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                (s₁ + s₂) =
                                              (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  s₁ +
                                                (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  s₂) }
                                    s) })
                    x +
                  (lift
                      {
                        toAddHom :=
                          {
                            toFun := fun s =>
                              {
                                toAddHom :=
                                  { toFun := fun y => s • y,
                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                map_smul' :=
                                  (_ :
                                    ∀ (r : R) (y : ↑Y),
                                      AddHom.toFun
                                          { toFun := fun y => s • y,
                                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                          (r • y) =
                                        (RingHom.id R) r •
                                          AddHom.toFun
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                            y) },
                            map_add' :=
                              (_ :
                                ∀ (s₁ s₂ : S),
                                  (fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) })
                                      (s₁ + s₂) =
                                    (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        s₁ +
                                      (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        s₂) },
                        map_smul' :=
                          (_ :
                            ∀ (r : R) (s : S),
                              AddHom.toFun
                                  {
                                    toFun := fun s =>
                                      {
                                        toAddHom :=
                                          { toFun := fun y => s • y,
                                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                        map_smul' :=
                                          (_ :
                                            ∀ (r : R) (y : ↑Y),
                                              AddHom.toFun
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                  (r • y) =
                                                (RingHom.id R) r •
                                                  AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    y) },
                                    map_add' :=
                                      (_ :
                                        ∀ (s₁ s₂ : S),
                                          (fun s =>
                                                {
                                                  toAddHom :=
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                  map_smul' :=
                                                    (_ :
                                                      ∀ (r : R) (y : ↑Y),
                                                        AddHom.toFun
                                                            { toFun := fun y => s • y,
                                                              map_add' :=
                                                                (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                            (r • y) =
                                                          (RingHom.id R) r •
                                                            AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              y) })
                                              (s₁ + s₂) =
                                            (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                s₁ +
                                              (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                s₂) }
                                  (r • s) =
                                (RingHom.id R) r •
                                  AddHom.toFun
                                    {
                                      toFun := fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) },
                                      map_add' :=
                                        (_ :
                                          ∀ (s₁ s₂ : S),
                                            (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                (s₁ + s₂) =
                                              (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  s₁ +
                                                (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  s₂) }
                                    s) })
                    y) }
      (s • z) =
    (RingHom.id S) s •
      AddHom.toFun
        {
          toFun :=
            ⇑(lift
                {
                  toAddHom :=
                    {
                      toFun := fun s =>
                        {
                          toAddHom :=
                            { toFun := fun y => s • y,
                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                          map_smul' :=
                            (_ :
                              ∀ (r : R) (y : ↑Y),
                                AddHom.toFun
                                    { toFun := fun y => s • y,
                                      map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                    (r • y) =
                                  (RingHom.id R) r •
                                    AddHom.toFun
                                      { toFun := fun y => s • y,
                                        map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                      y) },
                      map_add' :=
                        (_ :
                          ∀ (s₁ s₂ : S),
                            (fun s =>
                                  {
                                    toAddHom :=
                                      { toFun := fun y => s • y,
                                        map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                    map_smul' :=
                                      (_ :
                                        ∀ (r : R) (y : ↑Y),
                                          AddHom.toFun
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                              (r • y) =
                                            (RingHom.id R) r •
                                              AddHom.toFun
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                y) })
                                (s₁ + s₂) =
                              (fun s =>
                                    {
                                      toAddHom :=
                                        { toFun := fun y => s • y,
                                          map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                      map_smul' :=
                                        (_ :
                                          ∀ (r : R) (y : ↑Y),
                                            AddHom.toFun
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                (r • y) =
                                              (RingHom.id R) r •
                                                AddHom.toFun
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                  y) })
                                  s₁ +
                                (fun s =>
                                    {
                                      toAddHom :=
                                        { toFun := fun y => s • y,
                                          map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                      map_smul' :=
                                        (_ :
                                          ∀ (r : R) (y : ↑Y),
                                            AddHom.toFun
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                (r • y) =
                                              (RingHom.id R) r •
                                                AddHom.toFun
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                  y) })
                                  s₂) },
                  map_smul' :=
                    (_ :
                      ∀ (r : R) (s : S),
                        AddHom.toFun
                            {
                              toFun := fun s =>
                                {
                                  toAddHom :=
                                    { toFun := fun y => s • y,
                                      map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                  map_smul' :=
                                    (_ :
                                      ∀ (r : R) (y : ↑Y),
                                        AddHom.toFun
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                            (r • y) =
                                          (RingHom.id R) r •
                                            AddHom.toFun
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                              y) },
                              map_add' :=
                                (_ :
                                  ∀ (s₁ s₂ : S),
                                    (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        (s₁ + s₂) =
                                      (fun s =>
                                            {
                                              toAddHom :=
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (y : ↑Y),
                                                    AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        (r • y) =
                                                      (RingHom.id R) r •
                                                        AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          y) })
                                          s₁ +
                                        (fun s =>
                                            {
                                              toAddHom :=
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (y : ↑Y),
                                                    AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        (r • y) =
                                                      (RingHom.id R) r •
                                                        AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          y) })
                                          s₂) }
                            (r • s) =
                          (RingHom.id R) r •
                            AddHom.toFun
                              {
                                toFun := fun s =>
                                  {
                                    toAddHom :=
                                      { toFun := fun y => s • y,
                                        map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                    map_smul' :=
                                      (_ :
                                        ∀ (r : R) (y : ↑Y),
                                          AddHom.toFun
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                              (r • y) =
                                            (RingHom.id R) r •
                                              AddHom.toFun
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                y) },
                                map_add' :=
                                  (_ :
                                    ∀ (s₁ s₂ : S),
                                      (fun s =>
                                            {
                                              toAddHom :=
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (y : ↑Y),
                                                    AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        (r • y) =
                                                      (RingHom.id R) r •
                                                        AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          y) })
                                          (s₁ + s₂) =
                                        (fun s =>
                                              {
                                                toAddHom :=
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                map_smul' :=
                                                  (_ :
                                                    ∀ (r : R) (y : ↑Y),
                                                      AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          (r • y) =
                                                        (RingHom.id R) r •
                                                          AddHom.toFun
                                                            { toFun := fun y => s • y,
                                                              map_add' :=
                                                                (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                            y) })
                                            s₁ +
                                          (fun s =>
                                              {
                                                toAddHom :=
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                map_smul' :=
                                                  (_ :
                                                    ∀ (r : R) (y : ↑Y),
                                                      AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          (r • y) =
                                                        (RingHom.id R) r •
                                                          AddHom.toFun
                                                            { toFun := fun y => s • y,
                                                              map_add' :=
                                                                (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                            y) })
                                            s₂) }
                              s) }),
          map_add' :=
            (_ :
              ∀ (x y : ↑((restrictScalars f ⋙ extendScalars f).obj Y)),
                (lift
                      {
                        toAddHom :=
                          {
                            toFun := fun s =>
                              {
                                toAddHom :=
                                  { toFun := fun y => s • y,
                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                map_smul' :=
                                  (_ :
                                    ∀ (r : R) (y : ↑Y),
                                      AddHom.toFun
                                          { toFun := fun y => s • y,
                                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                          (r • y) =
                                        (RingHom.id R) r •
                                          AddHom.toFun
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                            y) },
                            map_add' :=
                              (_ :
                                ∀ (s₁ s₂ : S),
                                  (fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) })
                                      (s₁ + s₂) =
                                    (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        s₁ +
                                      (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        s₂) },
                        map_smul' :=
                          (_ :
                            ∀ (r : R) (s : S),
                              AddHom.toFun
                                  {
                                    toFun := fun s =>
                                      {
                                        toAddHom :=
                                          { toFun := fun y => s • y,
                                            map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                        map_smul' :=
                                          (_ :
                                            ∀ (r : R) (y : ↑Y),
                                              AddHom.toFun
                                                  { toFun := fun y => s • y,
                                                    map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                  (r • y) =
                                                (RingHom.id R) r •
                                                  AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    y) },
                                    map_add' :=
                                      (_ :
                                        ∀ (s₁ s₂ : S),
                                          (fun s =>
                                                {
                                                  toAddHom :=
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                  map_smul' :=
                                                    (_ :
                                                      ∀ (r : R) (y : ↑Y),
                                                        AddHom.toFun
                                                            { toFun := fun y => s • y,
                                                              map_add' :=
                                                                (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                            (r • y) =
                                                          (RingHom.id R) r •
                                                            AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              y) })
                                              (s₁ + s₂) =
                                            (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                s₁ +
                                              (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                s₂) }
                                  (r • s) =
                                (RingHom.id R) r •
                                  AddHom.toFun
                                    {
                                      toFun := fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) },
                                      map_add' :=
                                        (_ :
                                          ∀ (s₁ s₂ : S),
                                            (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                (s₁ + s₂) =
                                              (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  s₁ +
                                                (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  s₂) }
                                    s) })
                    (x + y) =
                  (lift
                        {
                          toAddHom :=
                            {
                              toFun := fun s =>
                                {
                                  toAddHom :=
                                    { toFun := fun y => s • y,
                                      map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                  map_smul' :=
                                    (_ :
                                      ∀ (r : R) (y : ↑Y),
                                        AddHom.toFun
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                            (r • y) =
                                          (RingHom.id R) r •
                                            AddHom.toFun
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                              y) },
                              map_add' :=
                                (_ :
                                  ∀ (s₁ s₂ : S),
                                    (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        (s₁ + s₂) =
                                      (fun s =>
                                            {
                                              toAddHom :=
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (y : ↑Y),
                                                    AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        (r • y) =
                                                      (RingHom.id R) r •
                                                        AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          y) })
                                          s₁ +
                                        (fun s =>
                                            {
                                              toAddHom :=
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (y : ↑Y),
                                                    AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        (r • y) =
                                                      (RingHom.id R) r •
                                                        AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          y) })
                                          s₂) },
                          map_smul' :=
                            (_ :
                              ∀ (r : R) (s : S),
                                AddHom.toFun
                                    {
                                      toFun := fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) },
                                      map_add' :=
                                        (_ :
                                          ∀ (s₁ s₂ : S),
                                            (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                (s₁ + s₂) =
                                              (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  s₁ +
                                                (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  s₂) }
                                    (r • s) =
                                  (RingHom.id R) r •
                                    AddHom.toFun
                                      {
                                        toFun := fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) },
                                        map_add' :=
                                          (_ :
                                            ∀ (s₁ s₂ : S),
                                              (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  (s₁ + s₂) =
                                                (fun s =>
                                                      {
                                                        toAddHom :=
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                        map_smul' :=
                                                          (_ :
                                                            ∀ (r : R) (y : ↑Y),
                                                              AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  (r • y) =
                                                                (RingHom.id R) r •
                                                                  AddHom.toFun
                                                                    { toFun := fun y => s • y,
                                                                      map_add' :=
                                                                        (_ :
                                                                          ∀ (b₁ b₂ : ↑Y),
                                                                            s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                    y) })
                                                    s₁ +
                                                  (fun s =>
                                                      {
                                                        toAddHom :=
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                        map_smul' :=
                                                          (_ :
                                                            ∀ (r : R) (y : ↑Y),
                                                              AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  (r • y) =
                                                                (RingHom.id R) r •
                                                                  AddHom.toFun
                                                                    { toFun := fun y => s • y,
                                                                      map_add' :=
                                                                        (_ :
                                                                          ∀ (b₁ b₂ : ↑Y),
                                                                            s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                    y) })
                                                    s₂) }
                                      s) })
                      x +
                    (lift
                        {
                          toAddHom :=
                            {
                              toFun := fun s =>
                                {
                                  toAddHom :=
                                    { toFun := fun y => s • y,
                                      map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                  map_smul' :=
                                    (_ :
                                      ∀ (r : R) (y : ↑Y),
                                        AddHom.toFun
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                            (r • y) =
                                          (RingHom.id R) r •
                                            AddHom.toFun
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                              y) },
                              map_add' :=
                                (_ :
                                  ∀ (s₁ s₂ : S),
                                    (fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) })
                                        (s₁ + s₂) =
                                      (fun s =>
                                            {
                                              toAddHom :=
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (y : ↑Y),
                                                    AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        (r • y) =
                                                      (RingHom.id R) r •
                                                        AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          y) })
                                          s₁ +
                                        (fun s =>
                                            {
                                              toAddHom :=
                                                { toFun := fun y => s • y,
                                                  map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (y : ↑Y),
                                                    AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        (r • y) =
                                                      (RingHom.id R) r •
                                                        AddHom.toFun
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                          y) })
                                          s₂) },
                          map_smul' :=
                            (_ :
                              ∀ (r : R) (s : S),
                                AddHom.toFun
                                    {
                                      toFun := fun s =>
                                        {
                                          toAddHom :=
                                            { toFun := fun y => s • y,
                                              map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (y : ↑Y),
                                                AddHom.toFun
                                                    { toFun := fun y => s • y,
                                                      map_add' :=
                                                        (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                    (r • y) =
                                                  (RingHom.id R) r •
                                                    AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      y) },
                                      map_add' :=
                                        (_ :
                                          ∀ (s₁ s₂ : S),
                                            (fun s =>
                                                  {
                                                    toAddHom :=
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                    map_smul' :=
                                                      (_ :
                                                        ∀ (r : R) (y : ↑Y),
                                                          AddHom.toFun
                                                              { toFun := fun y => s • y,
                                                                map_add' :=
                                                                  (_ :
                                                                    ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                              (r • y) =
                                                            (RingHom.id R) r •
                                                              AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                y) })
                                                (s₁ + s₂) =
                                              (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  s₁ +
                                                (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  s₂) }
                                    (r • s) =
                                  (RingHom.id R) r •
                                    AddHom.toFun
                                      {
                                        toFun := fun s =>
                                          {
                                            toAddHom :=
                                              { toFun := fun y => s • y,
                                                map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (y : ↑Y),
                                                  AddHom.toFun
                                                      { toFun := fun y => s • y,
                                                        map_add' :=
                                                          (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                      (r • y) =
                                                    (RingHom.id R) r •
                                                      AddHom.toFun
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                        y) },
                                        map_add' :=
                                          (_ :
                                            ∀ (s₁ s₂ : S),
                                              (fun s =>
                                                    {
                                                      toAddHom :=
                                                        { toFun := fun y => s • y,
                                                          map_add' :=
                                                            (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                      map_smul' :=
                                                        (_ :
                                                          ∀ (r : R) (y : ↑Y),
                                                            AddHom.toFun
                                                                { toFun := fun y => s • y,
                                                                  map_add' :=
                                                                    (_ :
                                                                      ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                (r • y) =
                                                              (RingHom.id R) r •
                                                                AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  y) })
                                                  (s₁ + s₂) =
                                                (fun s =>
                                                      {
                                                        toAddHom :=
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                        map_smul' :=
                                                          (_ :
                                                            ∀ (r : R) (y : ↑Y),
                                                              AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  (r • y) =
                                                                (RingHom.id R) r •
                                                                  AddHom.toFun
                                                                    { toFun := fun y => s • y,
                                                                      map_add' :=
                                                                        (_ :
                                                                          ∀ (b₁ b₂ : ↑Y),
                                                                            s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                    y) })
                                                    s₁ +
                                                  (fun s =>
                                                      {
                                                        toAddHom :=
                                                          { toFun := fun y => s • y,
                                                            map_add' :=
                                                              (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) },
                                                        map_smul' :=
                                                          (_ :
                                                            ∀ (r : R) (y : ↑Y),
                                                              AddHom.toFun
                                                                  { toFun := fun y => s • y,
                                                                    map_add' :=
                                                                      (_ :
                                                                        ∀ (b₁ b₂ : ↑Y),
                                                                          s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                  (r • y) =
                                                                (RingHom.id R) r •
                                                                  AddHom.toFun
                                                                    { toFun := fun y => s • y,
                                                                      map_add' :=
                                                                        (_ :
                                                                          ∀ (b₁ b₂ : ↑Y),
                                                                            s • (b₁ + b₂) = s • b₁ + s • b₂) }
                                                                    y) })
                                                    s₂) }
                                      s) })
                      y) }
        z

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