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Can't do it. The engine and transmission cases are different from 2002 to 2007. You would have to buy a cassette kit or complete transmission to upgrade your 2002 to a 6 speed.
Take it to a shop, or if you prefer to do it yourself then find a website that teaches you how to.
2007 for the FLSTF Fatboy had a 6 speed transmission
2006 Harley Davidson introduced the Dyna motorcycles with a 6 speed transmission, and in 2007 the 6 speed was added across the board.
2007
32 ounces
changing oil on 2007 roadglide
It came out in on the Dyna glide in 2006, all other models had it in 2007.
If I am not mistaken, all 2007 Harley Davidson's were fuel injected and would not have a carburetor.
2007
2007
There isn't one.