Eri and Deku are two of the most precious cinnamon rolls in all of My Hero Academia; Eri alone can overload MHA fans' hearts with her cuteness. They both have this sort of kindness and innocence that makes people want nothing to happen to either of them.
Part of that might have to do with how the two met: Eri was being horribly mistreated by the yakuza, and Deku made it his sole purpose to save her. Since then, seeing Eri find happiness in day-to-day life has become a great source of joy among MHA fans, especially when Deku is involved. In light of that, here's everything there is to know about the most wholesome relationship in My Hero Academia.
How Did Eri and Deku Meet?
Eri was initially in the custody of the Shie Hassaikai, a once-powerful yakuza group looking to regain its former glory. Its leader and Eri's "father," Chisaki Kai or "Overhaul," had been using her blood to manufacture anti-Quirk bullets and even performed inhuman experiments on her. She eventually found her way to the outside world and into Deku's warm embrace, her first real experience with kindness, but she was pulled back into the darkness by Overhaul's threats and verbal abuse. Deku, realizing how wrong this was, regretted not saving her then and there and made it his personal mission to do so next time.
In an intense fight between the Yakuza and the Heroes, Eri realized the only way to stop the violence was to let herself be saved, so she leaped back into Deku's arms. Deku vowed to never let Eri go again and defeated Overhaul. Eri was finally saved.
Does Eri Live With Deku?
As cute as it is to see Eri and Deku together, the two of them do not live in the same place. Eri's Quirk, Rewind, allows her to reverse a person's state of being; if she's not careful, she could reverse someone to a state of nonexistence. Due to the volatile nature of her Quirk, she is instead watched over by Aizawa Shōta, whose own Quirk, Erasure can cancel out her Quirk before it gets out of hand.
Eri lives on the UA campus in the Faculty Dorms. From here, she can be carefully monitored by Aizawa and the rest of the Pro Heroes on staff. She also regularly interacts with the students at UA who have gotten to know her, especially Deku and the Big Three. One could argue that Deku lives with Eri in the sense that they both live on UA's campus, just in separate Dorms.
Does Eri Heal Deku?
Eri's Rewind Quirk, for how dangerous it can be, also has the potential for great healing purposes. As Eri is able to reverse a living target back to a previous state, she can reverse a person's injuries to make it as if they were never hurt in the first place. An extreme example of this came up during Deku's fight with Overhaul. As it turned out, Deku was able to constantly fight at 100%, something that would normally destroy his body, as long as Eri was clinging to him and rewinding him. He also didn't run the risk of being rewound into nothingness because he was destroying his body too fast for Eri to rewind it that far. Unfortunately, Deku cannot rely on Eri's Rewind to heal him. She hasn't got enough control over it to use it reliably. If Deku needs healing, he'll have to do it the old-fashioned way.
What Does Eri Call Deku?
Eri refers to Deku by his Hero name. When Deku initially visited Eri in the hospital after the confrontation with the Shie Hassaikai, Eri wasn't sure what to call her savior. Deku initially introduced himself by his real name, Midoriya Izuku, but then he insisted on being called by his Hero name. He felt as though Deku would be easier to remember. As a result, Eri regularly refers to Deku by his Hero name rather than his civilian name. She also extends this gesture to Lemillion, another Hero who was greatly responsible for her rescue. She recognizes these people best as the Heroes they are, so that's how she refers to them.