Proctor Teagan is the quartermaster on the Prydwen. He is in charge of managing the Brotherhood's equipment and munitions and cataloging weapons and armor recovered by Proctor Quinlan's research patrols out in the field. As the Brotherhood's Proctor for the Order of the Sword, Teagan's scribes are responsible for designing and manufacturing the weapons used by the military. He is also the man responsible for engineering Maxson's battlecoat.
He sells a variety of gear to the discerning customer, as long as they are a member of the Brotherhood of Steel. However, he keeps the best gear reserved for higher-ranking members, which the Sole Survivor will be able to purchase as they rise through the ranks of the Brotherhood. The Proctor also manages the Brotherhood's food supply, which is used by the airship's mess officer to cook the meals eaten by the soldiers. The player can go and help gather food for the Brotherhood by securing a supply of crops from a settlement in the Commonwealth.
Teagan is a friend of a Lancer named Rico, whose Vertibird is shot down by a missile launcher-wielding raider shortly before the player returns to the Prydwen, after destroying the Institute. He will relate this story to the player in their first time conversing after the destruction of the Institute, to convey the point that grave losses often occur in order to achieve great victories. Just as Rico gave him, he will give the player a bottle of whiskey as a token of appreciation.
ps4xboxonepc Proctor Teagan may not talk after interacting with him later in the game. This can be due to the Sole Survivor having started The Nuclear Option with the Minutemen. Once The Nuclear Option is finished and the Institute is dealt with, Proctor Teagan will interact with the Sole Survivor again. [verified]
On PC, one might be able to fix this with setstage BoSR05 255 but that only will complete his quest.