Disclaimer: I don't own One Piece.
Why? Why the hell did he have agreed to that? Was it a new secret technique? Was he insane? Those questions were examples about what everybody, who have heard about that ähm unusual agreement, was asking now. Not only to themselves, but also to the whole world. They just couldn't understand this. It was so stupid, ridiculous and erroneous. So different from his typical behaviour.
So yes, you could say the news that Mihawk Dulacre, who is better known as Hawkeye, who lives alone in a large castle on a lonely island, who is not what you say a very social or nice person, who can freeze the strongest man with one look, who is one of the Shichibukai, who is one of the dangerous men on the world, who is simply the greatest swordsman of the world, was going to train Roronoa Zoro, who wants to become the greatest swordsman of the world for his part.
It was suspect. Who in the world would help his rival and enemy to surpass oneself?
Mihawk was aware of these questions. He was aware that the people were talking behind his back about this. But it helps when you are one of the most dangerous men on the world. The people doesn't dare to talk disrespectful in his presence (except Shanks), only for the reason that they weren't suicidal (except Shanks).
But if Mihawk would have to be honest, he would have to admit that he asked himself the same questions. Why have he agreed to train Roronoa Zoro? He met many strong swordsmen, but he never even thought about training them. But Roronoa Zoro was special.
When he first saw him Mihawk wasn't impressed. That teen wasn't worth it that he fought serious with him. Maybe he was one of the best swordsmen in the East Blue, but that didn't say so much. Hawkeye has seen much better swordsmen. But there was something that he hasn't have seen in a long time. He possessed a unique strong will. Such a kind of a strong will that allowed his owner to go over every single rule which existent.
And then he knew it. Roronoa Zoro was the one who was not only worth enough to surpass him but also would become skilled enough for it.
That's why he didn't kill him back then.
He hoped that when they next meet each other Roronoa Zoro would be capable to surpass him.
They met earlier then Mihawk have expected.
Roronoa Zoro wasn't ready to beat him.
And that wasn't the only disappointment. He had the nerve to ask his enemy to train him while putting all of his pride away.
Mihawk couldn't understand it first. Had Roronoa Zoro lose all of his strong will? Then he realized it. No, he had not. He only was ready to give up his pride for someone else, but this only confirmed his strong will. He was ready to go to the end. For his dream and for the dream of his captain.
This was the moment when Mihawk knew what he had to do. He had to train Roronoa Zoro. Also for his sake.
One day he would die. Then it would be unimportant if he were undefeated. It would give a new person who would own the title 'Greatest swordsman of the world'. There was no way to stop this. The only thing he could do was to find someone worth who would be able to get this title, either after his dead or through surpassing him when he is still alive. He found this person, the one who was a worth successor, the future greatest swordsman of the world. Roronoa Zoro.
End