∀ {R : Type u_4} {S : Type u_5} {A : Type u_3} {B : Type u_2} {C : Type u_1} [inst : CommSemiring R]
  [inst_1 : CommSemiring S] [inst_2 : Algebra R S] [inst_3 : Semiring A] [inst_4 : Algebra R A] [inst_5 : Algebra S A]
  [inst_6 : IsScalarTower R S A] [inst_7 : Semiring B] [inst_8 : Algebra R B] [inst_9 : Semiring C]
  [inst_10 : Algebra S C] (f : TensorProduct R A B ≃ₗ[S] C)
  (h_mul : ∀ (a₁ a₂ : A) (b₁ b₂ : B), f ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) = f (a₁ ⊗ₜ[R] b₁) * f (a₂ ⊗ₜ[R] b₂))
  (h_one : f (1 ⊗ₜ[R] 1) = 1) (r : S),
  (↑↑(Algebra.TensorProduct.algHomOfLinearMapTensorProduct (↑f) h_mul h_one).toRingHom).toFun
      ((algebraMap S (TensorProduct R A B)) r) =
    (algebraMap S C) r