Corollary

Let E and F be semisimple modules of type $\mathscr{C}$. The module E is isomorphic to F if and only if we have $[E]_{\mathscr{C}}= [F]_{\mathscr{C}}$ in $K(\mathscr{C})$.

Indeed, we have $[E]_{\mathscr{C}}=\sum_{S\in\mathscr{S}}\ell_S(E)[S]_{\mathscr{C}}$ and an analogous formula for F, and E is isomorphic to F if and only if we have $\ell_S(E) =\ell_S(F)$ for every $S\in \mathscr{S}$ (VIII, p. 72).