Corollary 2

Tag 05KC, permanent. Cite it as [Bourbaki, Tag 05KC].

Algebra, chapter II, § 1. Modules

Let $(E_i)_{i \in I}, (E'_i)_{i \in I}, (F_\lambda)_{\lambda \in L}, (F'_\lambda)_{\lambda \in L}$ be four families of $A$ -modules and, for each $i \in I$ (resp. each $\lambda \in L$ ), $u_i : E'_i \to E_i$ (resp. $v_\lambda : F_\lambda \to F'_\lambda$ ) a linear mapping. Then the diagram

\begin{array}{ccc} \operatorname{Hom}\left( \bigoplus_{i \in I} E'_i, \prod_{\lambda \in L} F'_\lambda \right) & \xrightarrow{\phi'} & \prod_{(i, \lambda) \in I \times L} \operatorname{Hom}(E'_i, F'_\lambda) \\ \uparrow & & \uparrow \\ \operatorname{Hom}\left( \bigoplus_i u_i, \prod_\lambda v_\lambda \right) & & \prod \operatorname{Hom}(u_i, v_\lambda) \\ \operatorname{Hom}\left( \bigoplus_{i \in I} E_i, \prod_{\lambda \in L} F_\lambda \right) & \xrightarrow{\phi} & \prod_{(i, \lambda) \in I \times L} \operatorname{Hom}(E_i, F_\lambda) \end{array}

(where $\phi$ and $\phi'$ are the canonical isomorphisms defined in Corollary 1 to Proposition 6) is commutative.

The verification follows immediately from the definitions.

When all the $E_i$ are equal to the same $A$ -module $E$ , the direct sum $\bigoplus_{i \in I} E_i$ is also denoted by $E^{(I)}$ : its elements are the mappings of $I$ into $E$ with finite support. If, for all $i$ , $f_i$ is taken to be the identity mapping $E \to E$ , by Proposition 6, a linear mapping $E^{(I)} \to E$ is obtained, called codiagonal, which associates with every family $(x_i)_{i \in I}$ of elements of $E$ , of finite support, its sum $\sum_{i \in I} x_i$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Nothing.

Corollary 2

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