So, when do you have it and when do you lose it?
Usually is: This character gets +2 something . (You have composure if your honor bid is lower than an opponent's.)
So, you can get it after the duel (biding lower than your opponent) or when you bid for draw card. That also mean that after every duel (not only from participating character), or anyway after any bidding the Composure change?
You can even gain composure *during* the duel, that might factor in to the participant's skill. Composure is constantly checked, basically. As soon as dials are changed, composure can turn on or off. Basically, if you ever wonder if you have composure, you just look at the honor dials.
There are some times outside of duels and the draw phase that dials can change, too. The one that readily comes to mind is Maze of Illusion, where one player sets their dial to a number. That change lasts for the turn, so players can gain or lose composure then, too. Just be mindful of if effects change your honor bid vs changing the result on the dial. It'll say on the card specifically, but a lot of abilities that affect honor bids don't actually change the dial itself.
https://fiveringsdb.com/rules/reference#honor