Separable algebras in multitensor C${}^{\ast }$-categories are unitarizable

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