l3lab/ntp-mathlib huggingface dataset:
/-- Given `R`-module X and `S`-module Y, `S`-linear linear maps `(extendScalars f).obj X ⟶ Y`
bijectively correspond to `R`-linear maps `X ⟶ (restrictScalars f).obj Y`.
-/
@[simps symm_apply]
def homEquiv {X Y} :
    ((extendScalars f).obj X ⟶ Y) ≃ (X ⟶ (restrictScalars.{max v u₂,u₁,u₂} f).obj Y) where
  toFun := HomEquiv.toRestrictScalars.{u₁,u₂,v} f
  invFun := HomEquiv.fromExtendScalars.{u₁,u₂,v} f
  left_inv g := by
    letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
    apply LinearMap.ext; intro z
    induction' z using TensorProduct.induction_on with x s z1 z2 ih1 ih2
    · rw [map_zero, map_zero]
    · erw [TensorProduct.lift.tmul]
      simp only [LinearMap.coe_mk]
      change S at x
      dsimp
      erw [← LinearMap.map_smul, ExtendScalars.smul_tmul, mul_one x]
      rfl
    · rw [map_add, map_add, ih1, ih2]
  right_inv g := by
    letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
    

Proof state:
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
X : ModuleCat R
Y : ModuleCat S
g : X ⟶ (restrictScalars f).obj Y
m1 : Module R S := Module.compHom S f
m2 : Module R ↑Y := Module.compHom (↑Y) f
⊢ HomEquiv.toRestrictScalars f (HomEquiv.fromExtendScalars f g) = g

Next tactic:
apply LinearMap.ext
/-- Given `R`-module X and `S`-module Y, `S`-linear linear maps `(extendScalars f).obj X ⟶ Y`
bijectively correspond to `R`-linear maps `X ⟶ (restrictScalars f).obj Y`.
-/
@[simps symm_apply]
def homEquiv {X Y} :
    ((extendScalars f).obj X ⟶ Y) ≃ (X ⟶ (restrictScalars.{max v u₂,u₁,u₂} f).obj Y) where
  toFun := HomEquiv.toRestrictScalars.{u₁,u₂,v} f
  invFun := HomEquiv.fromExtendScalars.{u₁,u₂,v} f
  left_inv g := by
    letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
    apply LinearMap.ext; intro z
    induction' z using TensorProduct.induction_on with x s z1 z2 ih1 ih2
    · rw [map_zero, map_zero]
    · erw [TensorProduct.lift.tmul]
      simp only [LinearMap.coe_mk]
      change S at x
      dsimp
      erw [← LinearMap.map_smul, ExtendScalars.smul_tmul, mul_one x]
      rfl
    · rw [map_add, map_add, ih1, ih2]
  right_inv g := by
    letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
    apply LinearMap.ext; 

Proof state:
case h
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
X : ModuleCat R
Y : ModuleCat S
g : X ⟶ (restrictScalars f).obj Y
m1 : Module R S := Module.compHom S f
m2 : Module R ↑Y := Module.compHom (↑Y) f
⊢ ∀ (x : ↑X), (HomEquiv.toRestrictScalars f (HomEquiv.fromExtendScalars f g)) x = g x

Next tactic:
intro x
/-- Given `R`-module X and `S`-module Y, `S`-linear linear maps `(extendScalars f).obj X ⟶ Y`
bijectively correspond to `R`-linear maps `X ⟶ (restrictScalars f).obj Y`.
-/
@[simps symm_apply]
def homEquiv {X Y} :
    ((extendScalars f).obj X ⟶ Y) ≃ (X ⟶ (restrictScalars.{max v u₂,u₁,u₂} f).obj Y) where
  toFun := HomEquiv.toRestrictScalars.{u₁,u₂,v} f
  invFun := HomEquiv.fromExtendScalars.{u₁,u₂,v} f
  left_inv g := by
    letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
    apply LinearMap.ext; intro z
    induction' z using TensorProduct.induction_on with x s z1 z2 ih1 ih2
    · rw [map_zero, map_zero]
    · erw [TensorProduct.lift.tmul]
      simp only [LinearMap.coe_mk]
      change S at x
      dsimp
      erw [← LinearMap.map_smul, ExtendScalars.smul_tmul, mul_one x]
      rfl
    · rw [map_add, map_add, ih1, ih2]
  right_inv g := by
    letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
    apply LinearMap.ext; intro x
    

Proof state:
case h
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
X : ModuleCat R
Y : ModuleCat S
g : X ⟶ (restrictScalars f).obj Y
m1 : Module R S := Module.compHom S f
m2 : Module R ↑Y := Module.compHom (↑Y) f
x : ↑X
⊢ (HomEquiv.toRestrictScalars f (HomEquiv.fromExtendScalars f g)) x = g x

Next tactic:
rw [HomEquiv.toRestrictScalars_apply, HomEquiv.fromExtendScalars_apply, lift.tmul,
      LinearMap.coe_mk, LinearMap.coe_mk]
/-- Given `R`-module X and `S`-module Y, `S`-linear linear maps `(extendScalars f).obj X ⟶ Y`
bijectively correspond to `R`-linear maps `X ⟶ (restrictScalars f).obj Y`.
-/
@[simps symm_apply]
def homEquiv {X Y} :
    ((extendScalars f).obj X ⟶ Y) ≃ (X ⟶ (restrictScalars.{max v u₂,u₁,u₂} f).obj Y) where
  toFun := HomEquiv.toRestrictScalars.{u₁,u₂,v} f
  invFun := HomEquiv.fromExtendScalars.{u₁,u₂,v} f
  left_inv g := by
    letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
    apply LinearMap.ext; intro z
    induction' z using TensorProduct.induction_on with x s z1 z2 ih1 ih2
    · rw [map_zero, map_zero]
    · erw [TensorProduct.lift.tmul]
      simp only [LinearMap.coe_mk]
      change S at x
      dsimp
      erw [← LinearMap.map_smul, ExtendScalars.smul_tmul, mul_one x]
      rfl
    · rw [map_add, map_add, ih1, ih2]
  right_inv g := by
    letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
    apply LinearMap.ext; intro x
    rw [HomEquiv.toRestrictScalars_apply, HomEquiv.fromExtendScalars_apply, lift.tmul,
      LinearMap.coe_mk, LinearMap.coe_mk]
    

Proof state:
case h
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
X : ModuleCat R
Y : ModuleCat S
g : X ⟶ (restrictScalars f).obj Y
m1 : Module R S := Module.compHom S f
m2 : Module R ↑Y := Module.compHom (↑Y) f
x : ↑X
⊢ ({ toFun := fun s => HomEquiv.evalAt f s g,
            map_add' :=
              (_ :
                ∀ (s₁ s₂ : S),
                  (fun s => HomEquiv.evalAt f s g) (s₁ + s₂) =
                    (fun s => HomEquiv.evalAt f s g) s₁ + (fun s => HomEquiv.evalAt f s g) s₂) }
          1).toAddHom
      x =
    g x

Next tactic:
dsimp
/-- Given `R`-module X and `S`-module Y, `S`-linear linear maps `(extendScalars f).obj X ⟶ Y`
bijectively correspond to `R`-linear maps `X ⟶ (restrictScalars f).obj Y`.
-/
@[simps symm_apply]
def homEquiv {X Y} :
    ((extendScalars f).obj X ⟶ Y) ≃ (X ⟶ (restrictScalars.{max v u₂,u₁,u₂} f).obj Y) where
  toFun := HomEquiv.toRestrictScalars.{u₁,u₂,v} f
  invFun := HomEquiv.fromExtendScalars.{u₁,u₂,v} f
  left_inv g := by
    letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
    apply LinearMap.ext; intro z
    induction' z using TensorProduct.induction_on with x s z1 z2 ih1 ih2
    · rw [map_zero, map_zero]
    · erw [TensorProduct.lift.tmul]
      simp only [LinearMap.coe_mk]
      change S at x
      dsimp
      erw [← LinearMap.map_smul, ExtendScalars.smul_tmul, mul_one x]
      rfl
    · rw [map_add, map_add, ih1, ih2]
  right_inv g := by
    letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
    apply LinearMap.ext; intro x
    rw [HomEquiv.toRestrictScalars_apply, HomEquiv.fromExtendScalars_apply, lift.tmul,
      LinearMap.coe_mk, LinearMap.coe_mk]
    dsimp
    

Proof state:
case h
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
X : ModuleCat R
Y : ModuleCat S
g : X ⟶ (restrictScalars f).obj Y
m1 : Module R S := Module.compHom S f
m2 : Module R ↑Y := Module.compHom (↑Y) f
x : ↑X
⊢ 1 • g x = g x

Next tactic:
rw [one_smul]
/--
For any `R`-module X, there is a natural `R`-linear map from `X` to `X ⨂ S` by sending `x ↦ x ⊗ 1`
-/
-- @[simps] Porting note: not in normal form and not used
def Unit.map {X} : X ⟶ (extendScalars f ⋙ restrictScalars f).obj X where
  toFun x := (1 : S)⊗ₜ[R,f]x
  map_add' x x' := by 

Proof state:
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
X : ModuleCat R
x x' : ↑X
⊢ (fun x => 1 ⊗ₜ[R] x) (x + x') = (fun x => 1 ⊗ₜ[R] x) x + (fun x => 1 ⊗ₜ[R] x) x'

Next tactic:
dsimp
/--
For any `R`-module X, there is a natural `R`-linear map from `X` to `X ⨂ S` by sending `x ↦ x ⊗ 1`
-/
-- @[simps] Porting note: not in normal form and not used
def Unit.map {X} : X ⟶ (extendScalars f ⋙ restrictScalars f).obj X where
  toFun x := (1 : S)⊗ₜ[R,f]x
  map_add' x x' := by dsimp; 

Proof state:
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
X : ModuleCat R
x x' : ↑X
⊢ 1 ⊗ₜ[R] (x + x') = 1 ⊗ₜ[R] x + 1 ⊗ₜ[R] x'

Next tactic:
rw [TensorProduct.tmul_add]
/--
For any `R`-module X, there is a natural `R`-linear map from `X` to `X ⨂ S` by sending `x ↦ x ⊗ 1`
-/
-- @[simps] Porting note: not in normal form and not used
def Unit.map {X} : X ⟶ (extendScalars f ⋙ restrictScalars f).obj X where
  toFun x := (1 : S)⊗ₜ[R,f]x
  map_add' x x' := by dsimp; rw [TensorProduct.tmul_add]
  map_smul' r x := by
    

Proof state:
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
X : ModuleCat R
r : R
x : ↑X
⊢ AddHom.toFun
      { toFun := fun x => 1 ⊗ₜ[R] x,
        map_add' :=
          (_ : ∀ (x x' : ↑X), (fun x => 1 ⊗ₜ[R] x) (x + x') = (fun x => 1 ⊗ₜ[R] x) x + (fun x => 1 ⊗ₜ[R] x) x') }
      (r • x) =
    (RingHom.id R) r •
      AddHom.toFun
        { toFun := fun x => 1 ⊗ₜ[R] x,
          map_add' :=
            (_ : ∀ (x x' : ↑X), (fun x => 1 ⊗ₜ[R] x) (x + x') = (fun x => 1 ⊗ₜ[R] x) x + (fun x => 1 ⊗ₜ[R] x) x') }
        x

Next tactic:
letI m1 : Module R S := Module.compHom S f
/--
For any `R`-module X, there is a natural `R`-linear map from `X` to `X ⨂ S` by sending `x ↦ x ⊗ 1`
-/
-- @[simps] Porting note: not in normal form and not used
def Unit.map {X} : X ⟶ (extendScalars f ⋙ restrictScalars f).obj X where
  toFun x := (1 : S)⊗ₜ[R,f]x
  map_add' x x' := by dsimp; rw [TensorProduct.tmul_add]
  map_smul' r x := by
    letI m1 : Module R S := Module.compHom S f
    -- Porting note: used to be rfl
    

Proof state:
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
X : ModuleCat R
r : R
x : ↑X
m1 : Module R S := Module.compHom S f
⊢ AddHom.toFun
      { toFun := fun x => 1 ⊗ₜ[R] x,
        map_add' :=
          (_ : ∀ (x x' : ↑X), (fun x => 1 ⊗ₜ[R] x) (x + x') = (fun x => 1 ⊗ₜ[R] x) x + (fun x => 1 ⊗ₜ[R] x) x') }
      (r • x) =
    (RingHom.id R) r •
      AddHom.toFun
        { toFun := fun x => 1 ⊗ₜ[R] x,
          map_add' :=
            (_ : ∀ (x x' : ↑X), (fun x => 1 ⊗ₜ[R] x) (x + x') = (fun x => 1 ⊗ₜ[R] x) x + (fun x => 1 ⊗ₜ[R] x) x') }
        x

Next tactic:
dsimp
/--
For any `R`-module X, there is a natural `R`-linear map from `X` to `X ⨂ S` by sending `x ↦ x ⊗ 1`
-/
-- @[simps] Porting note: not in normal form and not used
def Unit.map {X} : X ⟶ (extendScalars f ⋙ restrictScalars f).obj X where
  toFun x := (1 : S)⊗ₜ[R,f]x
  map_add' x x' := by dsimp; rw [TensorProduct.tmul_add]
  map_smul' r x := by
    letI m1 : Module R S := Module.compHom S f
    -- Porting note: used to be rfl
    dsimp; 

Proof state:
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
X : ModuleCat R
r : R
x : ↑X
m1 : Module R S := Module.compHom S f
⊢ 1 ⊗ₜ[R] (r • x) = r • 1 ⊗ₜ[R] x

Next tactic:
rw [← TensorProduct.smul_tmul,TensorProduct.smul_tmul']
/-- 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
  

Proof state:
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
⊢ (restrictScalars f ⋙ extendScalars f).obj Y ⟶ Y

Next tactic:
letI m1 : Module R S := Module.compHom S f
/-- 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
  

Proof state:
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
⊢ (restrictScalars f ⋙ extendScalars f).obj Y ⟶ Y

Next tactic:
letI m2 : Module R Y := Module.compHom Y f
/-- 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
  

Proof state:
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
⊢ (restrictScalars f ⋙ extendScalars f).obj Y ⟶ Y

Next tactic:
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' := _}
/-- 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' := _}
  · 

Proof state:
case refine'_1
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
⊢ ∀ (r : R) (x : ↑Y),
    AddHom.toFun { toFun := fun y => s • y, map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) }
        (r • x) =
      (RingHom.id R) r •
        AddHom.toFun { toFun := fun y => s • y, map_add' := (_ : ∀ (b₁ b₂ : ↑Y), s • (b₁ + b₂) = s • b₁ + s • b₂) } x

Next tactic:
intros r y
/-- 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
    

Proof state:
case refine'_1
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
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

Next tactic:
dsimp
/-- 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
    

Proof state:
case refine'_1
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
r : R
y : ↑Y
⊢ s • r • y = r • s • y

Next tactic:
change s • f r • y = f r • s • y
/-- 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
    

Proof state:
case refine'_1
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
r : R
y : ↑Y
⊢ s • f r • y = f r • s • y

Next tactic:
rw [← mul_smul, mul_comm, mul_smul]
/-- 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]
  · 

Proof state:
case refine'_2
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 : 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) })
        (x + y) =
      (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) })
          x +
        (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) })
          y

Next tactic:
intros s₁ s₂
/-- 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
    

Proof state:
case refine'_2
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
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

Next tactic:
ext y