How did Peter come to agree with Paul at the Jerusalem conference?

Les explains that Paul went to Jerusalem and withstood them to the face, insisting there was no way he would ever teach his pagan converts recently come out of idolatry that they had to keep the Judaic Law or practice circumcision, because they were now saved by grace through faith plus nothing. They had this great conference, and Les imagines it went on a good part of the day until finally Peter, by the work of the Holy Spirit, woke up and realized that long before—twelve years earlier—God had sent him to the house of Cornelius and showed him that He could save a Gentile. So Peter said, “I guess Paul’s right,” and they gave in. That is Galatians chapter 2. Later, James, Cephas (Peter), and John—the inner core or “inner sanctum”—perceived the grace given to Paul and gave him and Barnabas the right hands of fellowship; they shook hands and agreed. Les notes this shows God changing the program out of Judaism and legalism into the age of grace.