For appropriate choices of dagger categories this restricts to various notions of self-adjointness:
a bounded self-adjoint linear operator on a Hilbert space is defined everywhere and equals its own adjoint. The self-adjoint operator in general, unbounded case, is however not precisely equal to its adjoint as their domains may differ. As unbounded operators do not form an algebra this notion is nevertheless not a special case of the above. See self-adjoint operator.