Corollary 1

Tag 05K8, permanent. Cite it as [Bourbaki, Tag 05K8].

Algebra, chapter II, § 1. Modules

Let $(E_i)_{i \in I}, (F_\lambda)_{\lambda \in L}$ be two families of A-modules. The mapping

\text{Hom}_A \left( \bigoplus_{i \in I} E_i, \prod_{\lambda \in L} F_\lambda \right) \to \prod_{(i, \lambda) \in I \times L} \text{Hom}_A(E_i, F_\lambda)

which associates with each $g \in \text{Hom}_A \left( \bigoplus_{i \in I} E_i, \prod_{\lambda \in L} F_\lambda \right)$ the family $(\operatorname{pr}_\lambda \circ g \circ j_i)$ , is a $\mathbf{Z}$ -module isomorphism (called canonical).

This follows from Proposition 6 and no. 5, Proposition 4.

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

Cited by

Proposition 3, § 4
Corollary 2, § 1
Proposition 10, § 2
Proposition 2, § 2
Proposition 3, § 2
Corollary 2, § 2
Lemma 3, § 4
Remark, § 7

Cites

Proposition 6