Corollary 1

Let $H$ be a $\mathbf{Z}$-module and $h : E \times F \to H$ a $\mathbf{Z}$-bilinear mapping satisfying conditions (1) and such that $H$ is generated by $h(E \times F)$. Suppose that for every $\mathbf{Z}$-module $G$ and every $\mathbf{Z}$-bilinear mapping $f$ of $E \times F$ into $G$ satisfying (1) there exists a $\mathbf{Z}$-linear mapping $g : H \to G$ such that $f = g \circ h$. Then, if $\phi$ denotes the canonical mapping of $E \times F$ into $E \otimes_A F$, there exists one and only one isomorphism $\theta$ of $E \otimes_A F$ onto $H$ such that $h = \theta \circ \phi$.

The hypothesis that $h(E \times F)$ generates $H$ implies the uniqueness of $g$; the corollary is then just the general uniqueness property of a solution of a universal mapping problem (Set Theory, IV, § 3, no. 1).