SPECULATION : HOW CAN THE DAWN RELATE TO CARROT ?
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Carrot lost her father/Brother Figure and Mentor after said Pedro choosed to take his life for Luffy in the belief that Luffy woudl bring the dawn of the world.
Carrot still doesn't know the meaning behind the concept of the dawn and she was surprised when Pedro said this:
This means that Carrot is currently probably wondering what is the Dawn of the world and what is the story behind the Mink tribe and that prophecy.
Its possible that the Duke talked to her about that OR ... not at all.
In reality, Carrot only needs to think about ONE word and her mind should go crazy because of its significance for Pedro!
Its fair to say that if Carrot hear the word "DAWN", she will want to know more. And the only place she should be able to understand that is on the sea. Perhaps even with the strawhats.
This is why we might be just at the start of Carrot's journey and why the Story of Carrot might be related to the Dawn.