How did Morgiana get her Household Vessel? Alibaba got Amon from a dungeon capturing a Djinn, yet Morgiana was able to get some sort of relative to Amon from within the dungeon of Zagan.

How does this work? I don't think they explained this anywhere.

Note: I haven't read any of the manga, which might be where the answers lie.
When Alibaba got Amon from a Dungeon, his sword became a Metal Vessel, because Amon moved inside Alibaba's sword after he completed the dungeon. In other words, Alibaba became a Dungeon Capturer.


  Dungeon Capturer (迷宮攻略者, Danjon Kōryakusha) is a person who
  “completes” or “clears” Dungeons. If they survive the Dungeon, most of
  the time that person acquires wealth and magical items, like a flying
  cloth or Djinn Metal Vessel.


But Morgiana has a Household Vessel. Because Morgiana isn't a Dungeon Capturer she is a Household Member. 


  Household Members are people who help a King Vessel capture Dungeons
  and forms a Household Vessel from that person's Djinn, or if they form
  a strong connection to the King Vessel. If the King Vessel does not
  have his/her Metal Vessel, the assistant can not use their Household
  Vessel


So, Morgiana was at the Dungeon when Alibaba got Amon and his Metal Vessel, then Morgiana made a connection to Alibaba's Amon. After this connection, was necessary an activation, and this only happened when they were at the dungeon of Zagan.

Here is the list of all Dungeon Capturer and their respective Household Members.