Why Mihawk will fight Kizaru:

As of chapter 956 the Shichibukai were disbanded and a force of 8 warships were sent to capture Dracule Mihawk.

Fuji says in 997 "let's just put our trust in SSG to do it's job" this is referring to them taking over as the next great power. Also in 997 Akainu says "we don't have the manpower to spare on Wano", we know Fuji & GB were preoccupied during reverie.

Leaving only one admiral unaccounted for, Kizaru. he was originally intending to go stop BM & Kaido until he was stopped by Akainu. And now he isn't an option to send to Wano, meaning he is preoccupied, most likely with the Shichibukai.

I suspect he was sent for Mihawk; we saw Vice Admiral Stainless for Buggy & we know Coby was sent for Hancock. We can tell theirs only fodder sent for Weevil, but it's kept hidden who was sent for Mihawk.

Kizaru Vs Mihawk works since in marineford he was portrayed as one of the four major powers who attempted to attack Whitebeard (among the three admirals) & Mihawk's whole aesthetic is Darkness and Kizaru's is Light.

Mihawk livess on Kuraigana Island, a dark and gloomy island, and wields Yoru (Night in English) & said Island's only other inhabitants are monkeys... While Kizaru has the light fruit & his moniker is literally yellow monkey.

Oh yeah and Mihawk is based on the vampire Dracula and a vampire's major weakness is sunlight, so there's something else.