Remark

(1) Proposition 1 shows that the ordered pair consisting of $E_{(B)}$ and $\phi_E$ is a solution of the universal mapping problem (Set Theory, IV, § 3, no. 1), where $\Sigma$ is the species of left B-module structures (the morphisms being B-linear mappings) and the $\alpha$-mappings are the A-linear mappings of E into a B-module.

(2) If E is an $(A, (C'_i); (D'_j))$-multimodule and F a $(B, (C''_h); (D''_k))$-multimodule, then the isomorphism (2) is linear with respect to the $((D'_j, (C''_h); (C'_i), (D''_k))$-multimodule structures of the two sides ($§ 1$, no. 14 and $§ 3$, no. 4).

(3) Let E be a left A-module, $a$ a two-sided ideal of A and $\rho : A \to A/a$ the canonical homomorphism. In the notation of $§ 3$, no. 6, Corollary 2 to Proposition 6, the A-module $E/aE$ is annihilated by $a$ and therefore has canonically a left $(A/a)$-module structure ($§ 1$, no. 12); it is immediate that the canonical mapping $\pi : \rho^*(E) \to E/aE$ defined in $§ 3$, no. 6, Corollary 2 to Proposition 6 is an isomorphism for the $(A/a)$-module structures.