Q. 1. Can you tell me what is the difference between Bishops, Coadjutors and Auxiliaries in the administration of a Diocese?
A. 1. As the Pope is the Head of the Catholic Church, the Bishop is the Head of a territory, such being called a Diocese.
Can. 381 §1. "A diocesan bishop in the diocese entrusted to him has all ordinary, proper, and immediate power which is required for the exercise of his pastoral function except for cases which the law or a decree of the Supreme Pontiff reserves to the supreme authority or to another ecclesiastical authority."
A Coadjutor Bishop is appointed to assist the Bishop as needed.
Can. 403 §3. "If it appears more opportune to the Holy See, it can appoint ex officio a coadjutor bishop who also has special faculties. A coadjutor bishop possesses the right of succession."
Should the diocesan Bishop retire or die, the Coadjutor Bishop immediately becomes his successor. There is no interruption in the management of the Diocese. The Coadjutor Bishop should already be familiar with the affairs of the Diocese.
Can. 409 §1. "When the episcopal see is vacant, the coadjutor bishop immediately becomes the bishop of the diocese for which he had been appointed provided that he has legitimately taken possession of it."
With the Auxiliary Bishop, it is different. He may be appointed to assist with the administration of a certain geographical section of the Diocese. Or it may involve only a certain aspect of it such as the Sacrament of Confirmation. In some Dioceses, the Auxiliary Bishop oversees the Catholic education within the Diocese, ensuring that Catholic Doctrines are taught at all levels, from Elementary Schools to the Universities.
Can. 403 §1. "When the pastoral needs of a diocese suggest it, one or more auxiliary bishops are to be appointed at the request of the diocesan bishop. An auxiliary bishop does not possess the right of succession."
Can. 403 §2. "In more serious circumstances, even of a personal nature, an auxiliary bishop provided with special faculties can be given to a diocesan bishop."
No matter their authority within the Diocese, the Coadjutor and Auxiliary Bishops are accountable to the Bishop. As such, they must work together at all time, ensuring that the mind of the Bishop is implemented in effort and intention. Such requires ongoing consultation on matters of major importance.
Can. 407 §1. "In order to foster the present and future good of the diocese as much as possible, a diocesan bishop, a coadjutor, and the auxiliary mentioned in ? can. 403, §2 are to consult one another on matters of major importance.
§2. In considering cases of major importance, especially of a pastoral character, a diocesan bishop is to wish to consult the auxiliary bishops before others.
§3. Since a coadjutor bishop and an auxiliary bishop are called to share in the solicitude of the diocesan bishop, they are to exercise their duties in such a way that they proceed in harmony with him in effort and intention."