"What are you doing holding my foot kid? Let go!" Says All Might "I can't let go now! We're too high up I would die!" After hearing this All Might lands on the nearest building.
"Kid, just shout and keep knocking on the door, i'm sure someone will let you down. I have to go now." "Wait, I have a question for you!" when All Might hears this he hesitates hoping he answer this kid's question before his time runs out. "Can I be a hero like you are? Keep the entire world safe, become a 'symbol of peace' and protecting everyone no matter who they are?"
After hearing this All Might hesitates to leave, but decides to leave quickly anyways because his time was almost up. As All Might was about to leave he suddenly transformed into a thin, sick looking man. "Eh? You're not All Might!"
"Sorry to tell you this kid, but I am All Might I was injured in a fight 5 years ago, it limits my ability to use my quirk, leaving me looking like this as my true form. I'm like one of the guys showing off at the beach, puffing my chest out to look like I used to do before my injuries. Keep this a secret, don't tell anyone or post it on the internet, it would cause chaos and panic."
"I promise not to tell anyone but was the fight 5 years ago against Toxic Chainsaw?" "Oh no, that guy was too weak to do this kind of thing to me, the person I told you about, called 'All For One' is who injured me this way. If you ever hear about him before becoming one of the strongest heroes, run."
"What? Is he really that powerful?" "Powerful enough to injure me in my prime leaving me like this? Yes" After saying this All Might turns and leaves Izuku contemplating what he was told, he walks down the building as well after All Might leaves, while moving in a daze.
Suddenly, he hears explosions and screaming, he runs to investigate the situation and finds Bakugo being attacked by the same slime villain that attacked him. 'Oh god its all my fault, if I hadn't held onto All Might he never would've escaped! I have to save Kacchan.'
He sprints through the crowd of heroes and bystanders, realizing no one else is doing anything and letting Kachhan die. "What are you heroes doing, my friends in trouble!" "Our quirks aren't compatible, we cant do anything against this villain" "So you're just going to sit there? Alright then I'll have to save him myself!"
Izuku remembers his new power, which he decided to call 'Frost Breath' and attempts to call it once more, remembering the pressure and cold feeling in his chest. He then blows with all his breath, aiming at the parts of the slime attached to the ground, taking away his mobility then aiming for the parts around Bakugo, he then punches the ice shattering it, sending the shards of ice into the slime villain and letting Bakugo escape the villain.
"Oh, its you again kid? thanks for the ammo!" as the slime villain says this he launches shards of ice and the green rocks at Izuku and the rest of the heroes, weakening the quirks of everyone in the vicinity but the slime villain. The villain then grabs both Bakugo and Izuku, deciding 2 bodies with strong quirks were better than one.
As the villain did this, everyone in the surroundings felt a change in the air pressure and the very weather, "Detroit Smash!" a booming scream is heard by everyone in the vicinity as clouds parted for the massive strength behind All Might's punch, defeating the villain completely. "Sorry kid, I was wrong, pro's are always risking their lives. With your desire and will to be a hero, to risk your life to save others, you can be a 'Symbol of Peace' as great or even better than I am."
At that moment, the entire crowd explodes with applause, clapping and cheering for the greatest hero, and his latest victory. The heroes praise Bakugo for his amazing explosion quirk and Izuku for his frost breath, superhuman strength, speed quirk combo. As well as Izuku's will to put himself at risk to save his classmate.
After everyone including Izuku and Bakugo left the scene, Bakugo finds Izuku to talk to him, "I know you're one of my greatest friends, but I didn't need your help there, I don't want to owe you anything." "I don't expect you to owe me anything, you're my friend, i'd save you no matter the cost and while you have a rough exterior; I know I can count on you to do the same." "As long as you know I didn't need your help Deku" "Yeah, yeah you tsundere" "What did you say? I'll kill you!".
As Izuku was walking home he was stopped by All Might, "Kid, you proved me wrong today, you showcased enough bravery and a will to sacrifice yourself most modern heroes lack severely. You stopped me from being a hypocrite and just watching the slime villain kill your classmate, because my hero time was up. All the greatest heroes have stories from their high school days, they all had one thing in common, their legs moved before they even thought about what they were doing. You had that feeling today didn't you?"
"How did you know? I saw Kachhan in trouble, his eyes looked like he was asking for help" "This is why I have found you a worthy inheritor of my quirk, with your own quirk providing you with enough physical strength to wield its power" "Inherit your quirk? isn't that impossible?"
"My quirk is special, it's a stockpiling quirk that can be transferred from one host to another, I am the eighth wielder of the quirk 'One For All' and I would like you, Izuku Midoriya, to inherit my quirk and become the next 'Symbol Of Peace'"
All might then takes a strand of his hair and holds it out to Izuku. "Eat this" "What? Is this a joke? Why do I have to eat one of your hairs that's gross" "My Quirk is transferred through DNA and willing the quirk to be given to the person ingesting the DNA, this is the least disgusting way for you to inherit it"
"Alright fine, but if this is a joke i'm never going to trust you again" he then takes the hair from All Might's hand and eats it quickly. "My Quirk will take a few hours to be inherited as you're still digesting the hair, meet me tomorrow at the Dagobah Municipal Beach, where I will train you to use your new quirk"