“Starting to root for this guy” as in “They sent me to kill her and I made another call” kind of root for this guy? Did he and Coulson make a deal after the whole whoops-Black-Widow-is-now-an-asset-sorry-Coulson deal where Coulson made Hawkeye promise to tell him when he started rooting for the person he was supposed to shoot in the head? How many times has this been a problem for Hawkeye? Is he, like, known in SHIELD for being the agent who might adopt the target you sent him to kill?
I firmly believe that this is what happened with him and Natasha. Like, Fury gives Clint the order, the next thing he knows Barton’s dragging Natasha in being like BOSS CAN I PLEASE KEEP HER I PROMISE TO CLEAN UP ANY BODY PARTS SHE LEAVES LYING AROUND PLEASE PLEASE PLEEEEEEEEASE