∀ (R : Type u_5) (S : Type u_4) (M₁ : Type u_1) (M₂ : Type u_2) (M₃ : Type u_3) [inst : CommSemiring R]
  [inst_1 : Semiring S] [inst_2 : AddCommMonoid M₁] [inst_3 : AddCommMonoid M₂] [inst_4 : AddCommMonoid M₃]
  [inst_5 : Algebra R S] [inst_6 : Module R M₁] [inst_7 : Module S M₁] [inst_8 : IsScalarTower R S M₁]
  [inst_9 : Module R M₂] [inst_10 : Module R M₃],
  TensorProduct.AlgebraTensorModule.lift
        (LinearMap.prodMapLinear R M₂ M₃ (TensorProduct R M₁ M₂) (TensorProduct R M₁ M₃) S ∘ₗ
          (TensorProduct.AlgebraTensorModule.mk R S M₁ M₂).prod (TensorProduct.AlgebraTensorModule.mk R S M₁ M₃)) ∘ₗ
      ((TensorProduct.AlgebraTensorModule.lTensor S M₁) (LinearMap.inl R M₂ M₃)).coprod
        ((TensorProduct.AlgebraTensorModule.lTensor S M₁) (LinearMap.inr R M₂ M₃)) =
    LinearMap.id