How did charlemagne relationship with the catholic church benefit both the emperor and the pope

Answer: pope gets Charlemagne's army to defend him while the church supports him with money

There was a mutual relationship between the Pope and Charlemagne, this was because the Charlemagne's army was defending the Pope while on the other hand Charlemagne had "God on his side". The Pope got the Papel States while Charlemagne had to go to heaven.