Coquasi-bialgebras with preantipode and rigid monoidal categories

By a theorem of Majid, every monoidal category with a neutral quasi-monoidal functor to finitely-generated and projective $\mathbb{k}$-modules gives rise to a coquasi-bialgebra. We prove that if the category is also rigid, then the associated coquasi-bialgebra admits a preantipode, providing in this way an analogue for coquasi-bialgebras of Ulbrich’s reconstruction theorem for Hopf algebras. When $\mathbb{k}$ is field, this allows us to characterize coquasi-Hopf algebras as well in terms of rigidity of finite-dimensional corepresentations.