Proposition 4

Let $E = \prod_{i \in I} E_i$ be the product of a family of A-modules $(E_i)_{i \in I}$. For every A-module F and every family of linear mappings $f_i : F \to E_i$ there exists one and only one mapping f of F into E such that $pr_i \circ f = f_i$ for all $i \in I$ and this mapping is linear.

This follows directly from the definitions.

Product of modules is "associative": if $(J_\lambda)_{\lambda \in L}$ is a partition of I, the canonical mapping

is an isomorphism.