= Solution
Conversely, if the <character module> $M^*$ is <injective>, every restriction map in the preceding display is surjective. The character <Tensor-hom adjunction> then makes the dual of $N'\otimes_AM\to N\otimes_AM$ surjective. Since $D$ detects injections, that tensor map is injective for every inclusion $N'\hookrightarrow N$. Hence $M$ is <flat>, and
$$
\boxed{(2)\Longrightarrow(1).}
$$
Together these two implications prove the <Lambek theorem>, with both the duality and the injective-cogenerator facts justified.
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