Remark 1

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

Let $f : M \to N$ be a homomorphism of a-modules. Using A, X, p. 103, prop. 2, one proves that the following diagrams are commutative:

\begin{array}{ccc} \mathrm{Tor}_{d-i}^A(M, H_A^d(A)) & \xrightarrow{\tau^i(M)} & H_A^i(M) \\ \downarrow \mathrm{Tor}_{d-i}(f,1) & & \downarrow H^i(f) \\ \mathrm{Tor}_{d-i}^A(N, H_A^d(A)) & \xrightarrow{\tau^i(N)} & H_A^i(N) \end{array}

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

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