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The arrow is the canonical multiplication map, not an arbitrary abstract module isomorphism. Its inclusion into $M$ is ${\mu }_I:I{\otimes }_{A}M\to M$. Dualizing identifies ${\mu }_I^{\ast }$ with the restriction ${\mathrm{Hom}}_A(A,M^{\ast })\to {\mathrm{Hom}}_A(I,M^{\ast })$. All these maps are surjective precisely when $M^{\ast }$ is injective, by the Baer criterion. The injective cogenerator of abelian groups detects injectivity of ${\mu }_I$, and the Lambek theorem gives flatness. No Noetherian or finite-generation hypothesis is needed.

Conversely, if the character module $M^{\ast }$ is injective, every restriction map in the preceding display is surjective. The character Tensor-hom adjunction then makes the dual of $N^{\prime }{\otimes }_{A}M\to N{\otimes }_{A}M$ surjective. Since $D$ detects injections, that tensor map is injective for every inclusion $N^{\prime }\hookrightarrow N$. Hence $M$ is flat, andTogether these two implications prove the Lambek theorem, with both the duality and the injective-cogenerator facts justified.