Corollary

Under the conditions of Proposition 5, (i),

and there are canonical isomorphisms

obtained by respectively associating with the class of an element $x = (x_i)$ of $\prod_i E_i$, mod $\operatorname{Ker}(f)$, (resp. with the class of an element $y = (y_i)$ of $\prod_i F_i$, mod. $\operatorname{Im}(f)$) the family of classes of the $x_i$ mod. $\operatorname{Ker}(f_i)$ (resp. the family of classes of the $y_i$ mod. $\operatorname{Im}(f)$).

In particular, for $f$ to be injective (resp. surjective, bijective, zero) it is necessary and sufficient that, for all $i \in I$, $f_i$ be injective (resp. surjective, bijective, zero).

If, for all $i \in I$, we consider a submodule $F_i$ of $E_i$, the module $\prod_{i \in I} F_i$ is a submodule of $\prod_{i \in I} E_i$ and by virtue of the Corollary to Proposition 5, there is a canonical isomorphism

An important example of a product of modules is that where all the factor modules are equal to the same module $F$; their product $F^I$ is then just the set of mappings of $I$ into $F$. The diagonal mapping $F \to F^I$ mapping $x \in F$ to the constant function equal to $x$ on $I$ is linear. If $(E_i)_{i \in I}$ is a family of $A$-modules and, for all $i \in I$, $f_i : F \to E_i$ is a linear mapping, then the linear mapping $x \mapsto (f_i(x))$ of $F$ into $\prod_{i \in I} E_i$ is the composition of the mapping $(x_i) \mapsto (f_i(x_i))$ of $F^I$ into $\prod_{i \in I} E_i$ and the diagonal mapping $F \to F^I$.