Carrying on Pedro's will; We now have Wanda directly stating on panel that Carrot has inherited Pedro's will. The context clues for this were pretty clear since chapter 877, but it was plainly spelled out in chapter 1056.
So, what did Pedro believe? What was his goal / will / dream?
Pedro wanted to join the crew of the Pirate King - but he was born too late to join Roger and too early to join Luffy.
Roger told Pedro he was on he waiting list and that every person has their time to shine. Pedro took those words to heart and formed the Nox Pirates for the purpose of gathering Poneglyphs, making ready the path for Roger's successor.
When Inu and Neko showed the Zou Road Poneglyph to Luffy, Pedro believed that the next Pirate King had at last arrived and he accompanied the Straw Hats to Whole Cake Island, knowing full well it would likely be his final journey, so that he could make sure they successfully retrieved the Road Poneglyph.
Then, in his final moments he imparted his will to Carrot and passed on his unwavering faith that Luffy is the one meant to succeed Roger.
He specified that the Straw Hats are meant to deliver the New Dawn to the world - fulfilling a long held prophecy for the Mink people.
Pedro's final words echoed the words Roger spoke to him when he was put on the waiting list for the Pirate King's crew: "Every person has their time to shine." He then added his own corollary, "Keep moving forward."
His will was to aid the Pirate King in fulfilling his destined purpose in delivering the New Dawn, both for the sake of the Mink people and for the world as a whole.
We know that on a global scale, this likely means liberation from the world's oppressors, Lord Imu and the World Government, but what does it specifically mean for the Minks? That part remains a mystery.
The Fishmen, Kozuki, and likely Giants all have analogous prophecies, which foretell the same future promised day. For the Fishmen, it means walking the path toward the sun. For the Kozuki, it means opening Wano's borders. But what about the Minks? How will their people be restored to their proper place in the world? The answer to that question can only be found on Laugh Tale.
So Carrot's motivating desire - to keep moving forward. Chasing Perospero to exact revenge for Pedro's death was a fool's errand from the start. Pedro sacrificed himself for Carrot and the rest of the crew to live, not to be killed avenging him. Carrot learned that lesson the hard way and now, thanks to Nekomamushi, is free from the need to attain vengeance.
So, what does it mean for Carrot to move forward in the way Pedro wanted? To rule as Queen of Zou? Maybe - but it doesn't seem o fit particularly well because Pedro wanted to join the Pirate King's crew. He believed that the way to fulfill the Mink's prophecy was to help Luffy become the Pirate King.
So, ultimately, Carrot's goal would be very similar to Jinbei's. We know that the Fishmen walking the path toward the sun is also contingent on Luffy discovering One Piece, just like the Mink's prophecy being fulfilled. Jinbei believes in Luffy in the way as did Pedro. And Pedro passed on that belief to Carrot.