Powers & Abilities Mihawk's epithet "Hawkeyes" has nothing to do with Observation haki.

#1
A lot of people like to tell me Mihawk has better observation haki than Shanks when he's featless, because of his own epithet.
Let's debunk this real quickly, Observation haki has nothing to do with your eyes, observation haki Is actually internal perception which actually does not recquire the use of the eyes, on the opposite, if you want to develop good CoO you must do training like this:

Fujitora has One of the best CoO of the series:

So yeah, stop this cap, Mihawk probably has that nickname because he probably has and acute vision, several times that of others characters, with a visual range of several degrees, but that has nothing to with CoO.
 

Gol D. Roger

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#10
CoO is basically taking your five senses to a next level.

Some people have a predominantly hearing-based CoO like Bepo,
some excel at precog like Kuri and Shanks,
some excel at sensing the presence like Fujitora,
some excel at eyesight like Usopp, etc.
 
#14
Mihawk showing Luffy the future if he attacks.
That's one of my favourite pages in the entire series. I believe Mihawk put on a bit of a threat with CoC and Luffy sub-consciously tapped into future sight halting his attack. Mihawk was surprised but satisfied, his goal was always to test and guide Luffy.

As for CoO Mihawk is bound to be top 2 in the series - him and Fujitora. There are several other impressive fighters with future sight but also prodigies like Shyarly and Aisa.

Coby, Luffy, Shanks, Rayleigh, Kaido, Aokiji, Katakuri, Enel are all good at it but Mihawk and Fujitora have been hyped the most.
 
#16
CoO Haki has Eye version as well, Usopp and Yasopp has it. Probably Augur has it as well.

But we see S-Hawk so he has the same eyes so the chances of Mihawk being from a different race and his eyes being special due to that is higher now.

Also Mihawk failed to very bad when he said Pre-TS Zolo can't beat Baboons, he was surprised to Zolo beating them. Compare that to Rayleigh sensing every animal on an entire island and tells how many of them are strong is definitely a higher level CoO than Mihawk's as well.

Perhaps due to his eye ability, either Mihawk developed CoO Eye version not the usual CoO, or he never even needed due to natural enhanced sight ability, thats also an option. Although Mihawk should have regular CoO at least. Big mom didn't need a good CoO as well (except for when sensing the power of Enma sword) and she is stronger than Mihawk.
 
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