I’d like to analyze and round out Mihawk as the 3 dimensional character he is. I’ll start explaining who Mihawk was before the war in Marineford.
Mihawk before that arc was a guy who was entirely self-centered. His title and his strength were entirely his own as shown by him being the only pirate in the world without a crew and living alone on a deserted island.
This is most emphasized by his relationship with Shanks. Mihawk as we saw with Zoro shows his respect to people by taking them more seriously in fights and in Shanks’ case, Mihawk doesn’t even see it worth the effort to draw his sword. And this is entirely self-centered since Mihawk doesn’t care why Shanks lost his arm to help Luffy, he just sees Shanks’ action as an affront to him and their rivalry.
This self-centered person would not have accepted Zoro’s show of throwing away his pride as a Swordsman by bowing to him. This is basically what Shanks did in Mihawk’s eyes. But something changed with Mihawk during the war, he gained perspective and in the end he listened to the reason Zoro was throwing away his pride unlike before where Mihawk didn’t listen or care to why Shanks lost his arm.
During the Marineford war two things happened. First, Mihawk entered the war with the aim of trying to pick a fight with the only man he believed was singularly worth fighting, Whitebeard and when he showed his hand, Jozu, stopped him. It begins to dawn on Mihawk that Whitebeard doesn’t need to be singularly strong like he always thought since he has allies.
Secondly, Mihawk encounters Luffy and once again, more and more of Luffy’s allies keep getting in Mihawk’s way. At this point, Mihawk says to himself that this ability of gathering allies is the greatest power he’s seen on the seas. But this isn’t necessarily a turning point for Mihawk since we already know that he is a very open minded character. He saw the execution of Roger and admitted to Luffy that Pirate King is a harder road than his own. Now he saw first hand that having allies prevented him from getting his fight with both Whitebeard and Luffy despite their difference in strength. The problem is though, Mihawk is self-centered and doesn’t care about any of these. These ambitions of Pirate King and power to get allies might be great things, but they don’t yet matter to Mihawk personally so he has gained some perspective but not changed yet.
In the post war arc is when the perspective he gained now starts challenging his self-centeredness. Zoro is the only character, as far as we know, who Mihawk has hope can grow to challenge him and yet Zoro is telling him fighting for the self-centered Swordsman pride that Mihawk does is no longer his motivation. Zoro now wants to fight and become the Strongest Swordsman in the World to help Luffy become Pirate King and for the first time these things are now becoming personal to Mihawk with added perspective. Mihawk finally listens and accepts Zoro’s plea given he’s seen that fighting for others is worthwhile in its own right since Luffy and Whitebeard fought for Ace and in turn very many people fought for them as well and even managed to stop him of all people.
Mihawk effectively changes, completing his character arc from a self-centered prick to a less self-centered prick (because people don’t just do 180s in their characters completely that easily). I believe the only way this can be considered a full character shift is when or if he accepts to fight Shanks again.