Definition 3

Let M be an isotypical A-module of type S. A description of M with respect to S is a pair $(V, \alpha )$, where V is a left vector space over the field D and $\alpha : S\otimes_DV\rightarrow M$ is an isomorphism of A-modules.

Every isotypical A-module M of type S has a canonical description: it is the pair (Hom$_A(S,M), \alpha_M$), where $\alpha_M: S\otimes_D$ Hom$_A(S,M)\rightarrow M$ is the isomorphism of A-modules characterized by $\alpha_M(s\otimes f) =f(s)$ (VIII, p. 62, Proposition 3, a)).