All Might doesn't exist

Aizawa has OFA - saves Izuku from sludge villain and immediately sees potential in him - Offers to train him in quirkless combat (He barely uses OFA) Izu accepts and after 10 months of training erasure surprises Izuku with the offering of OFA as long as Izuku doesn't use it as his main source of combat - Izuku agrees

OFA works a little differently: You get a boost to your own quirk (if you have one), you gain the quirk of your predecessor (In this case Izuku gets Aizawa's boosted Erasure), and the strength/stockpile increase of the quirk.

Aizawa's quirk's: Boosted erasure (when activated instead of looking at his targets it makes a field around him for about 25 feat that erases quirks (not mutations)), He had Nana's Float quirk, and the stockpile/strength.