Traces on Module Categories over Fusion Categories

We consider traces on module categories over pivotal fusion categories which are compatible with the module structure. It is shown that such module traces characterise the Morita classes of special haploid symmetric Frobenius algebras. Moreover, they are unique up to a scale factor and they equip the dual category with a pivotal structure. This implies that for each pivotal structure on a fusion category over the complex numbers there exists a conjugate pivotal structure defined by the canonical module trace.