Definition 2

Tag 0DX2, permanent. Cite it as [Bourbaki, Tag 0DX2].

Algebra, chapter X, § 5. Modules d’extensions

We call the composite homomorphism the connecting homomorphism of the extension modules relative to the module $M$ and to the exact sequence $\mathscr{E}$

\delta(M, \mathscr{E}) = \varphi_M(N') \circ \partial_{(M\mathscr{E})} \circ \varphi_M(N'')^{-1} : \operatorname{Ext}_A(M, N'') \to \operatorname{Ext}_A(M, N')

This is a graded $k$ -homomorphism of ascending degree 1, whose homogeneous components are denoted $\delta^n(M, \mathscr{E}) : \operatorname{Ext}_A^n(M, N'') \to \operatorname{Ext}_A^{n+1}(M, N')$ .

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

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