Example 3

Tag 0DZI, permanent. Cite it as [Bourbaki, Tag 0DZI].

Let $M, N, M', N'$ be four left $A$ -modules. If $u \in \mathrm{Ext}_A(M, N)$ , $f \in \mathrm{Hom}_A(M', M)$ , $g \in \mathrm{Hom}_A(N, N')$ , then

(3)

\mathrm{Ext}_A(f, g)(u) = g \circ u \circ f \in \mathrm{Ext}_A(M', N').

This provides a new proof of the $k$ -bilinearity of the mapping $(f, g) \to \mathrm{Ext}_A(f, g)$ (X, p. 88, Prop. 6).

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

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