这是一份umass麻省大学 MATH 101作业代写的成功案例
$$
\beta: A \times B \longrightarrow \operatorname{Hom}{T}\left(C, A \otimes{R}\left(B \otimes_{S} C\right)\right)
$$
is a bihomomorphism of bimodules, and there is a unique homomorphism
$$
\bar{\beta}: A \otimes_{R} B \longrightarrow \operatorname{Hom}{T}\left(C, A \otimes{R}\left(B \otimes_{S} C\right)\right)
$$
of bimodules such that $\bar{\beta}(a \otimes b)=\beta(a, b)$ for all $a, b$. By $5.6,(u, c) \longmapsto \bar{\beta}(u)(c)$ is a bihomomorphism of $\left(A \otimes_{R} B\right) \times C$ into $A \otimes_{R}\left(B \otimes_{S} C\right)$. Hence there is a bimodule homomorphism
$$
\theta:\left(A \otimes_{R} B\right) \otimes_{S} C \longrightarrow A \otimes_{R}\left(B \otimes_{S} C\right)
$$
MATH101 COURSE NOTES ：
Proof. The set Bihom $(A \times B, C)$ of all bihomomorphisms of $A \times B$ into $C$ is an abelian group under pointwise addition. The universal property of the tensor map $\tau:(a, b) \longmapsto a \otimes b$ provides a bijection $\varphi \longmapsto \varphi \circ \tau \circ \operatorname{Hom}{\mathbb{Z}}\left(A \otimes{R} B, C\right)$ onto Bihom $(A \times B, C)$, which preserves pointwise addition. Proposition $5.2$ provides two more bijections:
$\operatorname{Bihom}(A \times B, C) \longrightarrow \operatorname{Hom}{R}\left(A, \operatorname{Hom}{Z}(B, C)\right)$,
$\operatorname{Bihom}(A \times B, C) \longrightarrow \operatorname{Hom}{R}\left(B, \operatorname{Hom}{\mathcal{Z}}(A, C)\right)$,