Catherine, Duchess of Cambridge would be correct IF she were divorced as happened with Diana when she became Diana, Princess of Wales and Sarah who is Sarah, Duchess of York. That was their styling after their divorces
HRH Catherine, Duchess of Cambridge would be correct IF she were widowed. This is the standard way a widowed peeress is referred to officially although in the past TM George VI and Elizabeth II allowed the widows of the late Dukes of Kent and Gloucester to be referred to as TRH Princess Marina and Princess Alice respectively. Princess Marina, of course, was Princess Marina from birth but in the UK she was Princess Edward. Having set that precedent when the late Duke of Gloucester died Elizabeth allowed her aunt to use HRH Princess Alice even though she had never had that style in her own right. Using that precedent it is conceivable that a widowed Kate may be allowed to use HRH Princess Catherine, Duchess of Cambridge but that decision would only be made at the time she became a widow - assuming she hasn't a higher title to use.
As she is neither the correct styling is HRH The Duchess of Cambridge - no use of the name Catherine at all. If she is to use a name it would be as HRH Princess William. On her marriage she lost the right to use her own name in official documents until she is HM Queen Catherine.
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