WARNING: The following contains spoilers for My Hero Academia Chapter 337, "A Disposable Life” by Horikoshi Kohei, Caleb Cook, and John Hunt, available now in English from Viz Media.
Since Yuga Aoyama was revealed as the UA traitor, many fans have waited with bated breath to find out what his fate would be. Not surprisingly, Izuku Midoriya, along with the rest of Class 1A, doesn't seem to have given up on their classmate; They're offering Yuga an opportunity to redeem himself.
Even though his previous status as a mole within UA was an important job, All For One never particularly cared for Yuga or his family. To him, they were just another group of his pawns that he could mobilize in advantageous situations as he saw fit. Their successes were always welcome but even though Yuga's identity has been exposed, it doesn't look like losing his spy will affect All For One much.
To Class 1A on the other hand, Yuga is an irreplaceable part of their group. Ever since they first entered UA, they'd helped one another in their journey towards becoming Pro Heroes. They each had their own motivations for choosing that path but as they trained, lived, and overcame challenges together, they formed bonds with each other and truly became a team. It was this sense of familiarity that they'd shared that made it so hard for 1A to believe that Yuga was actually a traitor.
The bond 1A shared was quite real to Yuga as well. Spying on his classmates for All For One was never a task he relished. Yuga was initially Quirkless and received his Navel Laser from All For One in an agreement between his parents and the villain leader. Spying on UA was how he was helping his parents fulfill their end of the bargain. It hurt Yuga every time his classmates were hurt in attacks that were partly his fault but that was the only way to protect his parents from All For One's wrath.
Like everyone else in his class, Yuga had a genuine desire to be a hero before entering UA. All For One's task may have corrupted his purpose a bit, but he had several moments when his selfless desire shone through. During Class 1A's attempt at the Provisional Hero License Exam, he made himself an obvious target to their opponents so the class could regroup around him. And even more bravely, he struck Mr. Compress with his Navel Laser during the League of Villains' attack on their training camp, preventing them from kidnapping Fumikage Tokoyami along with Katsuki Bakugo. It was this instance that Izuku reminded Yuga of when the latter claimed he was nothing more than a rotten villain.
Izuku also referenced Lady Nagant, another hero who had found herself in All For One's employ. She didn't work for him because she believed in his ideals but because her spirit had been crushed and manipulated, almost exactly like Yuga. Instead of casting him out, as many would expect, Izuku stretched out a helping hand to Yuga. Izuku fully understands that a person's past actions do not dictate their entire character and is willing to give Yuga another chance to prove himself. And it looks like Yuga's going to take it.
