Another possibility is that in the world where Goku Black comes from, there is an enemy who follows Goku Black to Izuku's world. This enemy could be a powerful Saiyan or a being with a dark ki similar to Goku Black. They might see Izuku as a weakness in Goku Black's plan and try to eliminate him first, making them the main antagonist in the fanfiction.
Well, perhaps it could be about Izuku learning Goku Black's unique combat styles and techniques. Maybe Goku Black would push Izuku to his limits in training, helping him gain new powers and a more ruthless edge in battle.
In a 'Harry Potter trained by Goku' fanfiction, the main antagonist could be an evil wizard who has allied himself with dark forces from other dimensions. This wizard could see Harry, with his new skills from Goku's training, as a major threat to his plans of world domination. He might have powers that are a mix of dark magic and some sort of energy manipulation similar to ki. Harry would then have to use his combined magic and Goku - trained skills to take on this powerful and devious enemy.
It could be Zamasu. Since Goku Black was Zamasu in Goku's body in the original story, perhaps a more powerful and enraged Zamasu could be the main antagonist. He might be jealous of the new fused form's power and seek to destroy it.
A new character could be the main antagonist. For example, a powerful alien who has heard of Goku Black's reputation and wants to prove himself by defeating the reincarnated warrior. This alien could have unique powers that are a challenge for Goku Black to overcome, like the ability to nullify ki attacks.
It could be someone who wants to steal the knowledge Naruto gained from the Black Panther. Maybe a rogue ninja or a power - hungry villain from another land.
Another option for the main antagonist could be Beerus. As a God of Destruction, he wouldn't be happy with Zamasu and Goku Black causing chaos. Beerus is extremely powerful and has the authority to deal with threats to the universe. He might see their actions as a direct challenge to his power and order. If he gets involved, he could completely change the dynamic of the story, as his power level is on a whole different scale compared to most characters in the Dragon Ball universe.
It could be a villain who is after Beast Boy's powers. Since Beast Boy is getting trained by Black Panther and becoming stronger, this villain might see him as a threat. Maybe it's someone like Deathstroke who wants to stop Beast Boy from reaching his full potential under Black Panther's tutelage.
If Naruto were trained by Goku Black in fanfiction, it would be a wild combination. Goku Black is from a different universe with different power systems. He could introduce Naruto to concepts like ki, which might blend with Naruto's chakra in interesting ways. Naruto's physical abilities would likely skyrocket, and his combat strategies would be completely revolutionized. He might start using more acrobatic and flashy moves, like Goku Black does.
Maybe there's a villain group that targets Izuku specifically because of his money. They want to steal his wealth to fund their evil schemes. They might try to kidnap him or hack into his financial accounts. Izuku then has to use his wits as well as his money to outsmart them.
The Joker could be a main villain. He might see Izuku as a new plaything and try to break his spirit during his training with Batman.