Now that you have prepared Eulmore and the Night's Blessed, you can tell Beq Lugg that you are ready.
This quest is available if you have completed the quest In Defiance of Fate.
Starting the Quest
Talk to Ryne in the Crystarium to begin the quest.
Speak with Beq Lugg
Go east to where the orange arrow is pointing in the minimap. You can optionally use the Aetheryte to go to the Dossal Gate. Go up the stairs and talk to the Crystarium Gatekeep, and say Yes to go to the Ocular. Talk to Beq Lugg to proceed.
Speak with Varshahn
Teleport to Radz-at-Han and use the Aetheryte to go to the Meghaduta. Go northeast to the quest icon and talk to Varshahn to proceed. There will be a cutscene.
Speak with Livingway
Teleport to Sinus Lacrimarum, then go northeast to where the orange arrow is pointing in the minimap. Talk to Livingway to proceed. There will be a cutscene.
Speak with Zero
Talk to Zero to complete the quest. There will be a cutscene.
Next Main Scenario Quest
After you complete the quest, talk to Varshahn to accept the next quest: Down in the Dark. There will be a cutscene.