Basically yes. From the pilot’s perspective, ILS categories I, II and III are really about minimums and visibility requirements. The aircraft itself doesn’t know which is which. You set the approach in the APP page as the ILS for the runway you want, set the minimums and that’s basically it. Autoland functionality of the aircraft doesn’t really take anything else into account.
The details of why one particular runway has a Cat II or III available are a bit more technical. Here’s what one of the 777 FCOM’s I’ve got have to say:
