In this short note, we classify linear categorified open topological field theories in dimension two by pivotal Grothendieck-Verdier categories, a type of monoidal category equipped with a weak, not necessarily rigid duality. In combination with recently developed string-net techniques, this leads to a new description of the spaces of conformal blocks of Drinfeld centers Z ( C ) Z(\mathcal {C}) of pivotal finite tensor categories C \mathcal {C} in terms of the modular envelope of the cyclic associative operad. If C \mathcal {C} is unimodular, we prove that the space of conformal blocks inherits the structure of a module over the algebra of class functions of C \mathcal {C} for every free boundary component. As a further application, we prove that the sewing along a boundary circle for the modular functor for Z ( C ) Z(\mathcal {C}) can be decomposed into a sewing procedure along an interval and the application of the partial trace. Finally, we construct mapping class group representations from Grothendieck-Verdier categories that are not necessarily rigid and make precise how these generalize existing constructions.