I always play the same tees at the same course. Is my Handicap Index® "established" from that set?

Your Handicap Index^® is not tied to a particular set of tees. Every time a score is posted, it is converted into a Score Differential™ — a measure that normalizes your performance against a course of standard difficulty — so your results are always comparable across different courses and tee sets.

As a result, your Handicap Index^® is a portable number and each time you play, it is converted into a Course Handicap™ specific to the tees and course you are playing.

Source: Rule 5, Rules of Handicapping