-- 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
M₁ M₂ :
  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
m₁ : (↑(M₁.obj X) : Type u)
m₂ : (↑(M₂.obj X) : Type u)
⊢ Eq (α :=
    (↑((ModuleCat.restrictScalars (CommRingCat.Hom.hom (R.map f))).obj
          (MonoidalCategoryStruct.tensorObj (𝒞 := ModuleCat.moduleCategory (↑(R.obj Y) : Type u)) (M₁.obj Y)
            (M₂.obj Y))) :
      Type u))
    ((ModuleCat.Hom.hom ((tensorObj M₁ M₂).map f) :
          (a : (↑(M₁.obj X ⊗ M₂.obj X) : Type u)) →
            (↑((ModuleCat.restrictScalars (CommRingCat.Hom.hom (R.map f))).obj (M₁.obj Y ⊗ M₂.obj Y)) : Type u))
        (m₁ ⊗ₜ[(↑(R.obj X) : Type u)] m₂) :
      (↑((ModuleCat.restrictScalars (CommRingCat.Hom.hom (R.map f))).obj
            (MonoidalCategoryStruct.tensorObj (𝒞 := ModuleCat.moduleCategory (↑(R.obj Y) : Type u)) (M₁.obj Y)
              (M₂.obj Y))) :
        Type u))
    ((ConcreteCategory.hom (FC :=
              fun
                (x1 x2 :
                  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)) ↦
              LinearMap (R :=
                (↑(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))
                (RingHom.id
                  (↑(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))
                (↑x1 : Type u) (↑x2 : Type u))
              (CC := ModuleCat.carrier) (M₁.map f) :
            (a : (↑(M₁.obj X) : Type u)) →
              (↑((ModuleCat.restrictScalars (RingCat.Hom.hom ((R ⋙ forget₂ CommRingCat RingCat).map f))).obj
                    (M₁.obj Y)) :
                Type u))
          m₁ ⊗ₜ[(↑(R.obj Y) : Type u)]
        (ConcreteCategory.hom (FC :=
              fun
                (x1 x2 :
                  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)) ↦
              LinearMap (R :=
                (↑(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))
                (RingHom.id
                  (↑(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))
                (↑x1 : Type u) (↑x2 : Type u))
              (CC := ModuleCat.carrier) (M₂.map f) :
            (a : (↑(M₂.obj X) : Type u)) →
              (↑((ModuleCat.restrictScalars (RingCat.Hom.hom ((R ⋙ forget₂ CommRingCat RingCat).map f))).obj
                    (M₂.obj Y)) :
                Type u))
          m₂ :
      TensorProduct (↑(R.obj Y) : Type u)
        (↑(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)))
              (M₁.obj Y)) :
          Type u)
        (↑(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)))
              (M₂.obj Y)) :
          Type u))

Tactic:
rfl

Premises used: Quotient.liftOn', AddCon.liftOn, List.foldl, TensorProduct.tmul, AddCon.toQuotient, FreeAddMonoid.sumAux, rfl, ModuleCat.instModuleCarrierObjRestrictScalars._aux_1