Just realized something while re-reading Bnha manga concerning Asui Tsuyu’s nickname. Probably a lot of people already know that, but let’s go:
First, we have this here. After the fight Izuku vs Bakugou, Asui really shows up for the first time, instantly telling Izuku that he can call her “Tsuyu”. Of course, he doesn’t do that right away, shy as he is. So it goes on:
Each time Izuku calls her “Asui”, she corrects him to “Tsuyu”. Of course, she probably just likes her first name better, but… she only ever corrects Izuku, no one else.
And then, there is the Tsuyu special (Chapter 84.005). Here, we see this scene, Asui making friends with a girl that will later become one of her best friends:
The special is set BEFORE Asui comes to U.A., and based on her reaction I dare to believe that this here is the first time she has ever been called Tsuyu by her friends. So, basically, those who can call her Tsuyu are those she calls her friends.
And to confirm that, we have this here:
See? Only her friends are allowed to call her Tsuyu. Meaning:
Since Izuku’s first fight at U.A., Tsuyu had kind of decided that this boy would become one of her friends. He didn’t know it himself at that time, but Tsuyu knew, and apparently, that’s more than enough.