Just to clear things up:
A-team is ransom ans hitchcock who both had two dope mixtapes: Hardhood classics 1& 2 . They split up but been seen together and fuck with the battle rapper called "Tay Roc"
Joe Budden, Fabolous, Paul Cain n Dj clue as "their coach" around this time formed the "Triangle Offense" unofficially. Several songs together mostly remixes to other singles between 2002-2004.
Meanwhile Budden was doing some "mentoring" for an new and upcoming rapper called Stack Bundles would go onto do alot of tracks with Joey, Fab, and Ateam b4 he jumped to Dipset around late 05- early 06.
Joey felt some type of way over that and him n Jim Jones threw little shots at each other via mixtape vids seen on youtube or random songs in general.
DJ Clue was basically the glue that kept all of them together.
On the low during that time period, I fucked with their material more than Giunit or Dipset but they were never an official movement or marketed that way. Sorta underground hood music.
Great time in hip hop for me.
EDIT: To wrap it up tho, since it seems like Ransom may be mentoring battle rapper "Tay Roc" and Joe Budden been mentoring battle rapper "Tsu Surf" for a few years now, that may have been the common element that helped them squash the shit together officially. Its sorta been squashed unofficially for awhile niggas are too old for young boy beef.