Recently, the study of higher categories of topological defects in quantum field theory has gained significant attention due to their connection to categorical symmetries. These higher categories exhibit noteworthy additional structures, depending upon the specific theories and defects under consideration. For instance, in oriented 2-dimensional field theories, they organize into a pivotal bicategory. Currently, we lack a comprehensive framework to systematically describe these intricate structures. In my talk I will argue that the theory of higher dagger categories provides such a framework. I will focus on defects within fully extended topological field theories. Except in low dimensions the picture proposed here is highly conjectural. The talk is partially based on joint work in progress with Bruce Bartlett, Gio Ferrer, Brett Hungar, Theo Johnson-Freyd, Cameron Krulewski, Nivedita, Dave Penneys, David Reutter, Claudia Scheimbauer, Luuk Stehouwer, and Chetan Vuppulury.