The Shichibukai system has officially been abolished and Akainu has ordered the Admirals to capture the Warlords.
Their biggest targets are Weevil, Boa Hancock, and Mihawk.
Kizaru vs Weevil
Kizaru has history with Whitebeard and he is the first admiral we see made aware of Weevil’s latest actions. For those reasons I think Kizaru would request to pursue Weevil himself because he acknowledges how powerful Weevil is and could be curious about Weevil’s connection with Whitebeard. Additionally, If Weevil is on Whitebeard’s island this could hint at Marco and Kizaru crossing paths again.
Fujitora vs Boa Hancock
Fujitora’s blindness will come in handy with dealing with Hancock, but part of Oda’s joke would probably be that he got sent there simply because the other two Admirals had their hands busy. Now, I think Fujitora would be the best Admiral to go capture Hancock because I think it’d help him understand the state of the world better. Hancock could detail the cruelty of the World Government and explain how she felt forced to become a warlord to protect her island. As a result, Fujitora would feel directly responsible for what happens to the Kuja Tribe since he was the one who promoted the destruction of the warlord system
Green Bull vs Mihawk
Green Bull obviously requested to capture Boa Hancock since he is fond of beautiful women. Akainu would angrily deny him because he knows Green Bull wouldn’t be able to handle himself on that island. With no options left: Green Bull is ordered to capture Mihawk, I think this would serve as the perfect introduction for Green Bull because it will immediately establish him as a force to be reckon with, and his personality would shine since he can apparently go from lazy to serious when needed (pretty much a mirror of Mihawk’s personality).
*Theory by SluttyCabbage