Proposition 6

Keep the notation above, and suppose that the A-module M is semisimple. The mapping $u\mapsto h(u)$ from the right D-vector space Hom$_A(M,S)$ to the dual of the left D-vector space Hom$_A(S,M)$ is D-linear and bijective.

Let $u\in$ Hom$_A(M,S),v\in$ Hom$_A(S,M)$, and $d\in D$. We have

This proves that the mapping $h$ is D-linear. It is simply the mapping given by $u\mapsto$ Hom(1$_S, u$) from Hom$_A(M,S)$ to Hom$_D$(Hom$_A(S,M)$, Hom$_A(S,S)$). To prove that it is bijective, by Proposition 5, b) of VIII, p. 66, it suffices to treat the case when M is isotypical of type S; we can then apply the corollary of VIII, p. 65.