Mihawk once saw a strong rival in Shanks. Their battles were said to shake the whole of the Grand Line and Mihawk often sought Shanks out for duels. Things got worse when Shanks lost his arm; to Mihawk, even though he trained so much, if he wins, it wouldn’t be fair anymore.
That’s why he was so bored, wandering through the seas – no one was strong enough – until he found Zoro. He recognizes that Zoro has talent and ambition – Zoro could be the one to make him feel alive again. Mihawk is prepared to wait as long as it takes for Zoro to become strong enough to beat him. He has a desire to see Zoro surpass him after having witnessed Zoro’s incredible will and determination after their battle. To this end, Mihawk watches Zoro’s progression and is delighted when Zoro gains his first bounty.
Mihawk was impressed on how Zoro’s determination to help Luffy to become the Pirate King was so strong, that he even went to his rival to ask by the most humilliating way possible to train him. So Mihawk realized that Zoro, as proud of himself as he is, had put his crew as priority instead of his own goals and accepted to train him.
After two years of intense training with Dracule Mihawk, Zoro has grown significantly more powerful than before and his skills in all the three swords styles have grown so powerful that he was able to defeat the fishman Hody Jones while underwater with a single sword strike. Mihawk taught Zoro also how to imbue his swords with Haki.
Now there is only one swordsman who Zoro aims to defeat in order to become the world’s best swordsman and that is none other than Dracule Mihawk.