∀ {R : Type u_1} {M : Type u_4} {N : Type u_5} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]
  [inst_2 : AddCommMonoid N] [inst_3 : Module R M] [inst_4 : Module R N] {A : Type u_7} {B : Type u_8}
  [inst_5 : CommSemiring A] [inst_6 : Algebra R A] [inst_7 : Semiring B] [inst_8 : Algebra R B] [inst_9 : Algebra A B]
  [inst_10 : IsScalarTower R A B] (f : M →ₗ[R] N),
  LinearMap.baseChange B (LinearMap.baseChange A f) =
    ↑(TensorProduct.AlgebraTensorModule.cancelBaseChange R A B B N).symm ∘ₗ
      LinearMap.baseChange B f ∘ₗ ↑(TensorProduct.AlgebraTensorModule.cancelBaseChange R A B B M)