If both of you post scores from the same course on the same day it should update automatically. If it doesn't, you can go to your rivalries page and edit any of them as long as you both threw the course on that day.
If both of you post scores from the same course on the same day it should update automatically. If it doesn't, you can go to your rivalries page and edit any of them as long as you both threw the course on that day.