While the general picture given by Kindlekitten was the ideal which the Church wanted to operate with, and there were stark influences from the Church towards the princes (the waging of Crusades, Church-declared truces), the real deeds were more complex than that. In the Middle Age of Europe the Pope acted like any other prince, dealing with all the other princes, including infidels, to his own profit and to augment his own power. That could lead to interesting consequences, like
establishing the Papacy in a French city if it served the interests of the Pope.