He's been called a lot of names, but the one that stuck was Hawk.

Tony offers Bruce something he hasn't had in a long time and then slowly includes the rest of the Avengers.

From Still Officially Lost by pollyrepeat: “Barton has clearly imprinted on you,” Fury starts, and keeps talking right over Phil’s knee-jerk, horrified, “He has not,” to seal Phil’s doom by saying, “so I’ve decided that you’re going to take primary responsibility for him from now on.” “Oh, god,” Phil says, faintly, before he can stop himself. // Otherwise known as, SHIELD: The Early Years. (PODFIC!)

“Barton has clearly imprinted on you,” Fury starts, and keeps talking right over Phil’s knee-jerk, horrified, “He has not,” to seal Phil’s doom by saying, “so I’ve decided that you’re going to take primary responsibility for him from now on.” “Oh, god,” Phil says, faintly, before he can stop himself. // Otherwise known as, SHIELD: The Early Years.