It is not a recommended workflow to make changes to the spatial reference via the feature class/dataset properties > XY Coordinate System tab, as only the definition of the coordinate system is modified. Although this is possible to complete, it is important to note that this workflow does not project the data stored within dataset(s).
Completing this workflow can have certain consequences such as displaying the data in incorrect locations or producing unexpected alignment. More specifically, this could cause the feature dataset to be incorrectly marked as high precision, without the ability to revert back to the original low precision coordinate system. Information on this software defect can be found at the following knowledge base article: KB45156: "Switching the spatial reference of a low-precision feature dataset may result in a mixed-precision feature dataset".
Changing the coordinate system should be accomplished through use of the 'Project' (Data Management) geoprocessing tool; this ensures that the underlying data and how it is displayed utilize the same coordinate system, and helps limit any inaccuracies that could occur.
Changing the coordinate system via ArcCatalog should rarely be done, however adding a vertical coordinate system to a feature class which is already Z-enabled is one acceptable case.