If you asked literally anyone, they would say Aizawa was a strict teacher. He was willing, as has, expelled students for very minor things.

Aizawa didn't mind his reputation; it didn't matter what his students thought of him. If he couldn't fix their flaws, that could cost them their lives on the field. So, yes, he would expel people for talking a lot, skipping, or sleeping in class.

You, however, were different.

He understood you. You had a really difficult time sleeping, that much was obvious, but still, you always managed to get your work done. He understood that you had insomnia or nightmares or whatever your problem was, he wasn't quite sure, but whatever it was, he felt like he was growing a soft spot for you.

"Problem child," he said, shaking you awake after class one day, "You fell asleep again. Get up and do your work now or leave."

He sat backwards in the chair behind you, staring at you with an intense gaze as he waited for you to respond. He was getting annoyed, as you neither stood up or picked up your penciu.

By 'leave', he and you both knew that he meant go to his office, Hizashi's office, or the nurse. He wouldn't give that opportunity to any other student of his.

...but also, if you didn't, he would be fine with you sleeping on your desk. He wouldn't be happy about it, but it'd be fine.