Okay so I need some help finding this one deku fanfic as I didn't add it to my library and started reading another story.
The story starts off with an A/N about all the trigger warnings. The first chapter is how Midoriya is tired and has planned a su**ide at 12 AM, meanwhile Aizawa catches onto his odd behavior and taps at him before screaming his name to stir him from his thoughts before class started, and later catches onto the fact that Midoriya is never happy and decides to keep a close eye on him.
After finishing doing homework with his friends, it is finally time to go up to the roof. Midoriya takes off at 11:50 and gets to the roof, taking his last 10 minutes to do something I forgot what. He leaves his jacket and shoes by the rails, changing the wallpaper of his phone to 'open and read note' and disabling his password.
Aizawa's senses go off at 11:50 PM and when checking a tracker, he finds that Midoriya is on the roof of his dorm and so he rushes to get him. He makes it just in time to catch the falling teen, making the boy pass out. The tired hobo brings Midoriya to his own dorm room and waits for the boy to wake up.
After Midoriya wakes up, he is in a panic of the unfamiliar environment he is in, but after his teacher comes in he relaxes and goes on making an excuse that he was reading a book on the roof. After some talking, the teen spills that his mother is mistreating him, and so Aizawa checks and takes care of his wounds, all the while drinking coffee and hot chocolate.
I think the story has coffee in the name, and I remember that it had a pretty black cover.