Remark 1

Tag 0DB3, permanent. Cite it as [Bourbaki, Tag 0DB3].

Commutative Algebra, chapter X, § 8. Dualité des modules de longueur finie

For every homomorphism $f : N \to N'$ of $A$ -modules, let us denote by $D(f) : D(N') \to D(N)$ the homomorphism $\mathrm{Hom}(f, 1_J)$ . Let $u : M \to M'$ and $v : P \to P'$ be homomorphisms of $A$ -modules. Choose projective resolutions $(C, p)$ of $M$ and $(C', p')$ of $M'$ , and a morphism of complexes $\tilde{u} : C \to C'$ such that $p' \circ \tilde{u} = u \circ p$ (A, X, p. 49, Proposition 3). The diagram

\begin{array}{ccc} \mathrm{Homgr}_A(C' \otimes_A P', J) & \xrightarrow{\mu'} & \mathrm{Homgr}_A(C', \mathrm{Hom}_A(P', J)) \\ \downarrow \mathrm{Hom}(\tilde{u} \otimes v, 1_J) & & \downarrow \mathrm{Hom}(\tilde{u}, \mathrm{Hom}(v, 1)) \\ \mathrm{Homgr}_A(C \otimes_A P, J) & \xrightarrow{\mu} & \mathrm{Homgr}_A(C, \mathrm{Hom}_A(P, J)) \end{array}

where $\mu$ and $\mu'$ are the canonical homomorphisms, is commutative; one then deduces from A, X, p. 103, Proposition 2 a commutative diagram

\begin{array}{ccc} \mathrm{Ext}_A^i(M', D(P')) & \xrightarrow{\theta^i(M', P')} & D(\mathrm{Tor}_i^A(M', P')) \\ \downarrow \mathrm{Ext}^i(u, D(v)) & & \downarrow D(\mathrm{Tor}_i(u, v)) \\ \mathrm{Ext}_A^i(M, D(P)) & \xrightarrow{\theta^i(M, P)} & D(\mathrm{Tor}_i^A(M, P)) \end{array}

Let $w : P'' \to P$ be a homomorphism of $A$ -modules; one obtains analogously a commutative diagram

\begin{array}{ccc} \mathrm{Tor}_i^A(M, D(P)) & \xrightarrow{\rho_i(M, P)} & D(\mathrm{Ext}_A^i(M, P)) \\ \downarrow \mathrm{Tor}_i(u, D(w)) & & \downarrow D(\mathrm{Ext}^i(u, w)) \\ \mathrm{Tor}_i^A(M', D(P'')) & \xrightarrow{\rho_i(M', P'')} & D(\mathrm{Ext}_A^i(M', P'')) \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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