Plot hole: In this film, the Yeti is Sulley and Mike's co-worker in the Monsters, Inc. Mailroom. But in Monsters, Inc, when Sulley and Mike are banished, they apparently don't recognise him.
Plot hole: During the final scare event, Mike gets a perfect score because Sully broke the simulator to allow the 6th scarer to get a perfect score no matter what. After the contest, the simulation resets and Mike pretends to scare the child in the simulator and it registers a perfect score on the scoreboard for the 6th scarer when it should really have registered a score for the 1st scarer, because the simulator had been reset.
Answer: They said he could rejoin the fraternity as long as he proved he could be a great scarer, so his victory showed that he was good enough for them.