This quest is unlocked after completing "Black Bart's Gambit." To start it, simply sail up to Roberts, who is waiting for you on his new ship in the southern waters of Chinchorro.
Edward is forced to track down and attempt to kill his old friend, Benjamin Hornigold, to prevent him from finding the Observatory.
Sail to the location marked on the map, taking care not to get too close to the nearby British ships. Hornigold is aboard a small schooner. Use your spyglass to identify him, or simply sail close enough.
Ignore the other ships and focus solely on the schooner Hornigold is on. Benjamin drops plenty of barrels behind him, so it's best not to sail too close directly behind him. Try to slow him down whenever possible by shooting at his sails, fire your cannons as soon as he is in range, and shoot his weak points. Ramming him isn't a bad idea either.
Once you manage to deplete his health bar, a cutscene will trigger showing Hornigold's ship reaching a nearby island on its last legs, with his crew - including Hornigold - heading ashore.
Hornigold is at the top of the ruins in the southwestern part of the island. To get there, you must first reach the section of the ruins located in the south of the island, as the only path leads from there. There are several ways to reach the southern ruins - I'll describe one that doesn't require open combat with the soldiers guarding the island.
Go ashore and assassinate the vomiting soldier standing to the left of the stone arch.
Continue straight ahead and hide in the tall grass growing beneath the rocks.
You can get rid of the big guy patrolling this part of the island by whistling and luring him into the tall grass, or you can simply sneak past him. Your goal is to reach the entrance at the base of the ruins directly ahead of you.
Take out the soldier you encounter in the cave. He has his back to you the entire time, so it'll be a piece of cake.
Climb up the wall. Watch out for the soldier. You can sneak up on him, but it's best to hide behind the low wall and whistle to lure him over to you. If you kill him where he's standing, there's a chance the soldier patrolling the ruins further to the right might see what you're doing.
Climb up onto the ruins, but don't step onto the platform where two soldiers are talking. Instead, move slightly to the right so they don't see you, but not too far, then climb onto the roof of the small stone building next to them at the top of the ruins.
From its roof, you'll be able to perform a double Air Assassination on the two soldiers. Just make sure the soldier patrolling the ruins on the right doesn't see you.
Use the slanted palm tree, then the tree and the next slanted trunk to reach the elevated area where another section of the ruins is located, and hide in the tall grass.
Without leaving cover, take out the guards in this area. First, wait for the soldier patrolling the area. Lure him into the tall grass with a whistle and assassinate him.
You can sneak past the two soldiers standing next to each other, but it's risky, so it's better to take them out as well, one by one, by luring them into the tall grass.
Next, enter the ruins and head to the left. You'll find yourself below the area where Hornigold is standing at the top of the ruins.
Carefully climb a little higher. Hornigold will have his back to you, so take him out with a single well-placed strike. A cutscene will then begin, after which you'll receive a reward - The Benjamin's Sails - and the quest will be completed. You'll also unlock the next main quest - The Observatory.
