You split up to question people in Kugane in the hopes of learning Yugiri and Gosetsu's whereabouts.
Requirements
This quest is available if you have completed the quest By the Grace of Lord Lolorito.
Starting the Quest
Talk to Alphinaud in the East Aldenard Trading Company's headquarters in the Ruby Bazaar in Kugane to begin the quest.
Speak with Lyse
Go to the quest icon on the map and talk to Lyse.
View the Sketch
While standing within the orange circle on the map, use the Yugiri and Gosetsu Sketch from the Key Items section of your inventory, or use the icon to the right of the quest name in your list of active quests.
Hand the Sketch to Karaku
Go outside and use the nearby Aethernet Shard to go to the Kugane Aetheryte Plaza. Go north across the bridge and attune to the Aethernet Shard there, then go west and up the stairs. Turn right, go up the stairs, and attune to the Aethernet Shard there, then enter the building to the north. Talk to Karaku and hand over the Yugiri and Gosetsu Sketch. There will be a cutscene.
Hand the Sketch to Kotokaze
Go outside and use the Aethernet Shard to go to the Shiokaze Hostelry Aethernet Shard, or, if you did not attune to it, go to the Kugane Aetheryte Plaza and go west and attune to the Shiokaze Hostelry Aethernet Shard. Go to the quest icon in the map and talk to Kotokaze to hand over the Yugiri and Gosetsu Sketch. There will be a cutscene.
Show the Sketch to the Captains
Go up the stairs that spiral around the inside of the building until you reach the door in the south wall. Go through that door, then go south across the bridge. Go clockwise around the red building until you can go west along a stone walkway. Go to the west end of the walkway and talk to the Meticulous Merchant to hand over the Yugiri and Gosetsu Sketch.
Go east along the stone walkway and you will find some stairs downward. Go down, then turn around and go west. The path takes you under the stone walkway. After you have gone through the arch, turn left and go down the wooden stairs, onto the red platform. Talk to the Curt Captain to hand over the Yugiri and Gosetsu Sketch.
Go east up the wooden stairs, then turn left and talk to the Hannish Spice Trader to hand over the Yugiri and Gosetsu Sketch.
Speak with Lyse Again
Go east, and go up the stairs to the marketplace. Use the Aethernet Shard to go to the Shiokaze Hostelry Aethernet Shard, and go to the quest icon on the map. Talk to Lyse to complete the quest.
Next Main Scenario Quest
After you complete the quest, talk to Lyse to accept the next quest: It's Probably a Trap.