If Oda had any sense, he would have already put Carrot out to sea in search of the guy who would bring the Dawn which is the SHs.
Well, technically, this is just some stuff her mentor said to her, she had other things to think about at this time so its logical that she is not really caring about that stuff right now.
It's already fucked, he already wrote that Carrot inherits the Dawn dream from Pedro
There is no "dream" from Pedro and even less so some "dream" inherited by Carrot, its an invention from the fans. Luffy bringing the dawn of the world is a simple belief related to a knowledge Pedro seemed to have about the future of the Mink tribe and some unknown prophecy.
Carrot only inherited his will to fight for the crew in a specific time at a specific place. Now, she needs the incentive to go beyond.
BTW
When I'm talking about "a prophecy" Pedro seemed to know about, its possible that its not the prophecy Oden discovered on Raftel but something that one of the Duke learned about after returning from Zou.
Either that, or Oda PURPOSEFULLY left a hole in the story of the Duke that we will know about later.
I've been warning you guys for year about this: In the story, Oden did NOT talk about the Dawn to its retainers. In fact we know that they were not aware AT ALL about what Oden discovered because they asked him themselve what he knew:
So HOW COME is Pedro aware of the "DAWN", a concept from a prophecy that is only mentionned in there ???
So, either the Duke and Pedro are aware of ANOTHER prophecy about the dawn of the world
OR
Oda offscreened the entire explaination to the retainers which is unlikely for such a news
OR
Pedro was indeed talking about the prophecy of the journal of Oden and there is a hole left in the story of Wano that involves the Dukes.
Because to know about the Prophecy, one had to have the journal in their hand and there are still
TWO mysteries left in the story of Wano:
1 - How did the Dukes went from this:
To this:
AND
2 - Who the HECK protected the journal of Oden ?