-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Module
Goal:
R : Type u
inst : CommRing R
A : Mon (ModuleCat R)
hf :
  Eq (α :=
    (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u) →ₗ[R]
      (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u) →ₗ[R]
        (↑(Functor.obj (𝟭 (Mon (ModuleCat R))) A).X : Type u))
    (compr₂ₛₗ (R₄ := R) (M := (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u)) (N :=
        (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u)) (P :=
        (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u) ⊗[R]
          (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u))
        (Q := (↑(Functor.obj (𝟭 (Mon (ModuleCat R))) A).X : Type u)) (σ₁₄ := RingHom.id R) (σ₂₄ := RingHom.id R) (σ₃₄ :=
        RingHom.id R)
        (TensorProduct.mk R (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u)
          (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u))
        (Hom.hom (μ ≫ ↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ })) :
      (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u) →ₗ[R]
        (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u) →ₗ[R]
          (↑(Functor.obj (𝟭 (Mon (ModuleCat R))) A).X : Type u))
    (compr₂ₛₗ (R₄ := R) (M := (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u)) (N :=
        (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u)) (P :=
        (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u) ⊗[R]
          (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u))
        (Q := (↑(Functor.obj (𝟭 (Mon (ModuleCat R))) A).X : Type u)) (σ₁₄ := RingHom.id R) (σ₂₄ := RingHom.id R) (σ₃₄ :=
        RingHom.id R)
        (TensorProduct.mk R (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u)
          (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u))
        (Hom.hom
          (MonoidalCategoryStruct.tensorHom (↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ })
              (↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ }) ≫
            μ)) :
      (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u) →ₗ[R]
        (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u) →ₗ[R]
          (↑(Functor.obj (𝟭 (Mon (ModuleCat R))) A).X : Type u))
⊢ Eq.{u + 1} (α :=
    (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R)
            (Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X
            (Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X) :
        Type u) →ₗ[R]
      (↑(Functor.obj (𝟭 (Mon (ModuleCat R))) A).X : Type u))
    (Hom.hom (μ ≫ ↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ }))
    (Hom.hom
      (MonoidalCategoryStruct.tensorHom (↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ })
          (↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ }) ≫
        μ))

Tactic:
refine TensorProduct.ext ?_

Premises used: TensorProduct.ext