You can where WHATEVER shoes you want in comp.
The pair I borrowed were a size too big for me but they seemed to fit okay for the 5/6 sessions I used them.
Adidas fit is narrower than the Nikes.
Nikes are wider and are more comfy for my feet. I have always had a preference for Nike trainers though.
I have 4 pairs of shoes
Adidas weightlifting shoes from 1999, size 9
AdiStars from 04 Olympics size 9
Nike Romaleos, white, 2007, size 8.5
Nike Romaleos, grey, 2008, size 9.5