How exactly did the deal between the church and charlemagne benefit both?

Charlemagne and the Church supported each other many times.  Charlemagne led military campaigns against that threaten the Church.  Many of the enemies he defeated converted to the Christian faith. He also intervened on behalf of Pope Leo whom he helped restore back to Rome after the pope was mistreated by the Romans.  In return, Pope crowned him emperor on Christmas Day in 799.  He was supported by the clergy for his wish to strengthen the piousness and morality of his vassals.

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