Two procedures for extending topological or uniform space concepts to bitopological or quasi-uniform spaces are: (1) spanning subcategories or functors by suitable objects; (2) lifting epireflections. The main theorem relates Cauchy completions of functorial admissible (quasi-) uniformities to generalized compactness reflections. We discuss the non-unique extension of the realcompactness reflection to bitopological spaces and the resulting bitopological version of Shirota's theorem.