Proposition 2

Let $\rho : A \to B,\ \sigma : B \to C$ be ring homomorphisms. For every left A-module E, there exists one and only one C-homomorphism

mapping $1 \otimes (1 \otimes x)$ to $1 \otimes x$ for all $x \in E$ and this homomorphism is bijective.

The underlying $\mathbf{Z}$-modules of $\sigma^*(\rho^*(E))$ and $(\sigma \circ \rho)^*(E)$ are respectively $C \otimes_B (B \otimes_A E)$ and $C \otimes_A E$. There exists a canonical $\mathbf{Z}$-isomorphism $C \otimes_B (B \otimes_A E) \to (C \otimes_B B) \otimes_A E$ ($§ 3$, no. 8, Proposition 8), which is also a C-isomorphism for the left C-module structures on both sides. Moreover, the

C-module $C \otimes_B B$ is canonically identified with the C-module $C_s$ under the isomorphism which maps $\gamma \otimes \beta$ to $\gamma \sigma(\beta)$ (§ 3, no. 4, Proposition 4) and this isomorphism is also an isomorphism for the right A-module structure on $C \otimes_B B$ defined by $\rho$ and the right A-module structure on C defined by $\sigma \circ \rho$. Thus a canonical isomorphism

is obtained and, composing it with the isomorphism

defined earlier, the desired canonical isomorphism is obtained.

If $\phi, \phi'$ and $\phi''$ denote the canonical mappings $E \to \rho^*(E), \rho^*(E \to) \sigma^*(\rho^*(E))$ and $E \to (\sigma \circ \rho)^*(E)$, $\phi' \circ \phi$ is identified with $\phi''$ under the canonical isomorphism of Proposition 2.