-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : Mon (ModuleCat R)
⊢ 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))
have 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' := sorry, map_smul' := sorry })) :
    (↑(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' := sorry, map_smul' := sorry })
            (↟{ toFun := _root_.id, map_add' := sorry, map_smul' := sorry }) ≫
          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))
Tactic: refine TensorProduct.ext ?_