-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Monoidal
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
R : CategoryTheory.Functor.{v_1, u, u_1, u + 1} Cᵒᵖ CommRingCat
M :
  PresheafOfModules
    (R ⋙
      forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
        CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
        RingCat.carrier))
X Y : Cᵒᵖ
f : X ⟶ Y
hf :
  ∀ (m : (↑(M.obj X) : Type u)),
    Eq (α :=
      (↑(Functor.obj
            (ModuleCat.restrictScalars
              (RingCat.Hom.hom
                (CategoryTheory.Functor.map
                  (R ⋙
                    forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                      CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                      (CD := RingCat.carrier))
                  f)))
            ((Monoidal.tensorObj M
                  (unit
                    (R ⋙
                      forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                        CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                        (CD := RingCat.carrier)))).obj
              Y)) :
        Type u))
      ((ModuleCat.Hom.hom
              (M.map f ≫
                CategoryTheory.Functor.map
                  (ModuleCat.restrictScalars
                    (RingCat.Hom.hom
                      (CategoryTheory.Functor.map
                        (R ⋙
                          forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                            (CC := CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦
                            (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier))
                        f)))
                  (Iso.hom
                    ((fun (x : Cᵒᵖ) ↦
                        (Iso.symm (ρ_ (M.obj x)) :
                          M.obj x ≅
                            MonoidalCategoryStruct.tensorObj (𝒞 :=
                              ModuleCat.moduleCategory
                                (↑(Functor.obj
                                      (R ⋙
                                        forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦
                                          (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier) RingCat (FD :=
                                          fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
                                          RingCat.carrier))
                                      x) :
                                  Type u))
                              (M.obj x)
                              ((𝟙_
                                  (ModuleCat
                                    (↑(Functor.obj
                                          (R ⋙
                                            forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦
                                              (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier) RingCat
                                              (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
                                              RingCat.carrier))
                                          x) :
                                      Type u)))
                                (𝒞 :=
                                ModuleCat.moduleCategory
                                  (↑(Functor.obj
                                        (R ⋙
                                          forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦
                                            (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier) RingCat (FD :=
                                            fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
                                            RingCat.carrier))
                                        x) :
                                    Type u)))))
                      Y))) :
            (a : (↑(M.obj X) : Type u)) →
              (↑((ModuleCat.restrictScalars (RingCat.Hom.hom ((R ⋙ forget₂ CommRingCat RingCat).map f))).obj
                    ((Monoidal.tensorObj M (unit (R ⋙ forget₂ CommRingCat RingCat))).obj Y)) :
                Type u))
          m :
        (↑(Functor.obj
              (ModuleCat.restrictScalars
                (RingCat.Hom.hom
                  (CategoryTheory.Functor.map
                    (R ⋙
                      forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                        CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                        (CD := RingCat.carrier))
                    f)))
              ((Monoidal.tensorObj M
                    (unit
                      (R ⋙
                        forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                          CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                          (CD := RingCat.carrier)))).obj
                Y)) :
          Type u))
      ((ModuleCat.Hom.hom
              (Iso.hom
                  ((fun (x : Cᵒᵖ) ↦
                      (Iso.symm (ρ_ (M.obj x)) :
                        M.obj x ≅
                          MonoidalCategoryStruct.tensorObj (𝒞 :=
                            ModuleCat.moduleCategory
                              (↑(Functor.obj
                                    (R ⋙
                                      forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦
                                        (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier) RingCat (FD :=
                                        fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
                                        RingCat.carrier))
                                    x) :
                                Type u))
                            (M.obj x)
                            ((𝟙_
                                (ModuleCat
                                  (↑(Functor.obj
                                        (R ⋙
                                          forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦
                                            (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier) RingCat (FD :=
                                            fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
                                            RingCat.carrier))
                                        x) :
                                    Type u)))
                              (𝒞 :=
                              ModuleCat.moduleCategory
                                (↑(Functor.obj
                                      (R ⋙
                                        forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦
                                          (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier) RingCat (FD :=
                                          fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
                                          RingCat.carrier))
                                      x) :
                                  Type u)))))
                    X) ≫
                (Monoidal.tensorObj M
                      (unit
                        (R ⋙
                          forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                            (CC := CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦
                            (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)))).map
                  f) :
            (a : (↑(M.obj X) : Type u)) →
              (↑((ModuleCat.restrictScalars (RingCat.Hom.hom ((R ⋙ forget₂ CommRingCat RingCat).map f))).obj
                    ((Monoidal.tensorObj M (unit (R ⋙ forget₂ CommRingCat RingCat))).obj Y)) :
                Type u))
          m :
        (↑(Functor.obj
              (ModuleCat.restrictScalars
                (RingCat.Hom.hom
                  (CategoryTheory.Functor.map
                    (R ⋙
                      forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                        CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                        (CD := RingCat.carrier))
                    f)))
              ((Monoidal.tensorObj M
                    (unit
                      (R ⋙
                        forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                          CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                          (CD := RingCat.carrier)))).obj
                Y)) :
          Type u))
⊢ M.map f ≫
      CategoryTheory.Functor.map
        (ModuleCat.restrictScalars
          (RingCat.Hom.hom
            (CategoryTheory.Functor.map
              (R ⋙
                forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
                  RingCat.carrier))
              f)))
        (Iso.hom
          ((fun (x : Cᵒᵖ) ↦
              (Iso.symm (ρ_ (M.obj x)) :
                M.obj x ≅
                  MonoidalCategoryStruct.tensorObj (𝒞 :=
                    ModuleCat.moduleCategory
                      (↑(Functor.obj
                            (R ⋙
                              forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                                (CC := CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦
                                (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier))
                            x) :
                        Type u))
                    (M.obj x)
                    ((𝟙_
                        (ModuleCat
                          (↑(Functor.obj
                                (R ⋙
                                  forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦
                                    (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier) RingCat (FD :=
                                    fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier))
                                x) :
                            Type u)))
                      (𝒞 :=
                      ModuleCat.moduleCategory
                        (↑(Functor.obj
                              (R ⋙
                                forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦
                                  (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier) RingCat (FD :=
                                  fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier))
                              x) :
                          Type u)))))
            Y)) =
    Iso.hom
        ((fun (x : Cᵒᵖ) ↦
            (Iso.symm (ρ_ (M.obj x)) :
              M.obj x ≅
                MonoidalCategoryStruct.tensorObj (𝒞 :=
                  ModuleCat.moduleCategory
                    (↑(Functor.obj
                          (R ⋙
                            forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                              (CC := CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦
                              (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier))
                          x) :
                      Type u))
                  (M.obj x)
                  ((𝟙_
                      (ModuleCat
                        (↑(Functor.obj
                              (R ⋙
                                forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦
                                  (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier) RingCat (FD :=
                                  fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier))
                              x) :
                          Type u)))
                    (𝒞 :=
                    ModuleCat.moduleCategory
                      (↑(Functor.obj
                            (R ⋙
                              forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                                (CC := CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦
                                (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier))
                            x) :
                        Type u)))))
          X) ≫
      (Monoidal.tensorObj M
            (unit
              (R ⋙
                forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
                  RingCat.carrier)))).map
        f

Tactic:
ext m