The main way in which Charlemagne’s relationship with the Catholic Church benefited both parties is that the Church was able to give him lots of money to rule, and he was able to support the Church, which made for a relatively placid population.
Charlemagne supported the Church by expecting Catholic membership and the Church in return supported Charlemagne by allowing him to be a strong leader.
Charlemagne took control of many affairs that had been handled by the Church including appointment to the Church. The Church supported this despite it took some power away because Charlemagne expected the people to follow the Church. This meant growth and more money for the Church within Charlemagne's empire.
