A strict dagger category is a strict 1-category 𝒞 with an antiinvolutive functor † : 𝒞 → 𝒞op that is the identity on objects. I will explain a natural coherent version of dagger higher category. In the case of higher categories with lots of duals and adjoints, it is natural to ask for "unitary adjoints", and I will explain how such notion is also natural, and also selects a good definition of higher pivotality. A special case is the extended cobordism category: building on a construction of Freed and Hopkins, the cobordism category (extended or unextended) is naturally dagger (with unitary duals, if extended), but only when the tangential structure is stable. The stably-framed cobordism (∞,n)-category satisfies the unitary cobordism hypothesis: it is freely generated by the point among symmetric monoidal dagger (∞,n)-categories with unitary duals. In particular, whereas the non-dagger cobordism category with tangential structure H is a noninvertible refinement of MTH(n), the dagger structure makes it act more like a noninvertible refinement of MTH. This talk is based on arXiv:2403.01651 with many authors and on joint work in progress with Cameron Krulewski, Lukas Müller, and Luuk Stehouwer.