Fox’s theorem

Fox’s theorem says that the forgetful functor \(U \colon \mathbf{Cat} \to \mathbf{SymMon}\) from cartesian to symmetric monoidal categories has a right adjoint \(\mathbf{Comon} \colon \mathbf{SymMon} \to \mathbf{Cat}\) given by the category of cocommutative comonoids. Dually, commutative monoids have the same correspondence with cocartesian categories. A bicategory of relations is a cartesian bicategory where every object is a Frobenius monoid. What are other instances of this correspondence?