Separable algebras in multitensor C$ ^* $-categories are unitarizable

S. Carpi et al. (Comm. Math. Phys., 402 (2023), 169–212) proved that every connected (i.e., haploid) Frobenius algebra in a tensor C$ ^* $-category is unitarizable (i.e., isomorphic to a special C$ ^* $-Frobenius algebra). Building on this result, we extend it to the non-connected case by showing that an algebra in a multitensor C$ ^* $-category is unitarizable if and only if it is separable.