/-- 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
    

Proof state:
case refine'_4
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
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✝

Next tactic:
rw [map_add]