 Solution

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.