Exarch is the voice and/or hand of the Holy Father. Someone with great authority who gets sent to intermediate and solve problems, grant the Holy Father’s blessings, raise armies, make Kings kneel, etc. Whenever the Holy Father’s emissaries need to wield ultimate authority on behalf of the Holy Father, you send an Exarch.
If you’ve ever gone to the crusades in FoHB, you would’ve had dealings with an Exarch.