Corollary

Let M be a semisimple A-module. For every submodule N of M and every element $\lambda$ of $\mathscr{S}$, identify Hom$_A(S_{\lambda},N)$ with the $D_{\lambda}$-linear subspace of Hom$_A(S_{\lambda},M)$ consisting of the mappings with image contained in N.

a) The mapping $N\mapsto$ (Hom$_A(S_{\lambda},N)$)$_{\lambda\in\mathscr{S}}$ is a bijection from the set of A-submodules of M to the set of families $(W_{\lambda})_{\lambda\in\mathscr{S}}$ such that for every $\lambda \in \mathscr{S}$, $W_{\lambda}$ is a $D_{\lambda}$-linear subspace of Hom$_A(S_{\lambda},M)$.

b) The inverse bijection sends a family $(W_{\lambda})_{\lambda\in\mathscr{S}}$ to the A-submodule $\sum_{\lambda\in\mathscr{S}}\sum_{w\in W_{\lambda}}w(S_{\lambda})$ of M.

This is a reformulation of Proposition 8, c) applied to the canonical description of M.