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Actually, some of the temporary responsibilities, and the temporary "leader" of the see is the auxiliary bishop:

§2. When the episcopal see is vacant and unless competent authority has established otherwise, an auxiliary bishop preserves all and only those powers and faculties which he possessed as vicar general or episcopal vicar while the see was filled until a new bishop has taken possession of the see. If he has not been designated to the function of diocesan administrator, he is to exercise this same power, conferred by law, under the authority of the diocesan administrator who presides offer the governance of the diocese.

In the case of the Diocese of Rome, there are several auxiliary bishops (see list in the table to the right here). The highest of them are the Cardinal Vicar and the Vicar General.

However, in the special case of Rome, the functions of the diocese are reduced considerably during a vacancy. As this article says (based on this apostolic constitution of 1996):

According to Universi Dominici gregis, the government of the Holy See and the administration of the Catholic Church during sede vacante falls to the College of Cardinals, but in a very limited capacity. At the same time, all the heads of the departments of the Roman Curia "cease to exercise" their offices. The exceptions are the Cardinal Camerlengo, who is charged with managing the property of the Holy See, and the Major Penitentiary, who continues to exercise his normal role. If either has to do something which normally requires the assent of the Pope, he has to submit it to the College of Cardinals. Papal legates continue to exercise their diplomatic roles overseas, and both the Vicar General of Rome and the Vicar General for the Vatican City State continue to exercise their pastoral role during this period. The postal administration of the Vatican City State prepares and issues special postage stamps for use during this particular period, known as "sede vacante stamps".

Actually, some of the temporary responsibilities, and the temporary "leader" of the see is the auxiliary bishop:

§2. When the episcopal see is vacant and unless competent authority has established otherwise, an auxiliary bishop preserves all and only those powers and faculties which he possessed as vicar general or episcopal vicar while the see was filled until a new bishop has taken possession of the see. If he has not been designated to the function of diocesan administrator, he is to exercise this same power, conferred by law, under the authority of the diocesan administrator who presides offer the governance of the diocese.

In the case of the Diocese of Rome, there are several auxiliary bishops (see list in the table to the right here). The highest of them are the Cardinal Vicar and the Vicar General.

However, in the special case of Rome, the functions of the diocese are reduced considerably during a vacancy. As this article says (based on this apostolic constitution of 1996):

According to Universi Dominici gregis, the government of the Holy See and the administration of the Catholic Church during sede vacante falls to the College of Cardinals, but in a very limited capacity. At the same time, all the heads of the departments of the Roman Curia "cease to exercise" their offices. The exceptions are the Cardinal Camerlengo, who is charged with managing the property of the Holy See, and the Major Penitentiary, who continues to exercise his normal role. If either has to do something which normally requires the assent of the Pope, he has to submit it to the College of Cardinals. Papal legates continue to exercise their diplomatic roles overseas, and both the Vicar General of Rome and the Vicar General for the Vatican City State continue to exercise their pastoral role during this period. The postal administration of the Vatican City State prepares and issues special postage stamps for use during this particular period, known as "sede vacante stamps".

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No, because Benedict XVI is an emeritus bishop of Rome, not a coadjutor bishop of Rome.

As the Canon Law says, when a bishop retires from office, he becomes an emeritus bishop:

Can. 402 §1. A bishop whose resignation from office has been accepted retains the title of emeritus of his diocese and can retain a place of residence in that diocese if he so desires, unless in certain cases the Apostolic See provides otherwise because of special circumstances.

In the case of Benedict, he is a Pope Emeritus because he is also the Bishop of Rome, which holds the title of Pope.

When a see becomes vacant, the Canon Law states that the coadjunt bishop (if there is one), takes the temporary vacancy:

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.

Not all sees have one though:

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. [...]

§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.

Since the coadjunt bishop has the right of succession, but the Bishop of Rome is elected through a Papal Conclave, there cannot be a coadjunct Bishop of Rome. Therefore, if Pope Francis dies, the see is vacant. Benedict XVI will have no temporary official leadership at all.

No, because Benedict XVI is an emeritus bishop of Rome, not a coadjutor bishop of Rome.

As the Canon Law says, when a bishop retires from office, he becomes an emeritus bishop:

Can. 402 §1. A bishop whose resignation from office has been accepted retains the title of emeritus of his diocese and can retain a place of residence in that diocese if he so desires, unless in certain cases the Apostolic See provides otherwise because of special circumstances.

In the case of Benedict, he is a Pope Emeritus because he is also the Bishop of Rome, which holds the title of Pope.

When a see becomes vacant, the Canon Law states that the coadjunt bishop (if there is one), takes the temporary vacancy:

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.

No, because Benedict XVI is an emeritus bishop of Rome, not a coadjutor bishop of Rome.

As the Canon Law says, when a bishop retires from office, he becomes an emeritus bishop:

Can. 402 §1. A bishop whose resignation from office has been accepted retains the title of emeritus of his diocese and can retain a place of residence in that diocese if he so desires, unless in certain cases the Apostolic See provides otherwise because of special circumstances.

In the case of Benedict, he is a Pope Emeritus because he is also the Bishop of Rome, which holds the title of Pope.

When a see becomes vacant, the Canon Law states that the coadjunt bishop (if there is one), takes the temporary vacancy:

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.

Not all sees have one though:

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. [...]

§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.

Since the coadjunt bishop has the right of succession, but the Bishop of Rome is elected through a Papal Conclave, there cannot be a coadjunct Bishop of Rome. Therefore, if Pope Francis dies, the see is vacant. Benedict XVI will have no temporary official leadership at all.

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No, because Benedict XVI is an emeritus bishop of Rome, not a coadjutor bishop of Rome.

As the Canon Law says, when a bishop retires from office, he becomes an emeritus bishop:

Can. 402 §1. A bishop whose resignation from office has been accepted retains the title of emeritus of his diocese and can retain a place of residence in that diocese if he so desires, unless in certain cases the Apostolic See provides otherwise because of special circumstances.

In the case of Benedict, he is a Pope Emeritus because he is also the Bishop of Rome, which holds the title of Pope.

When a see becomes vacant, the Canon Law states that the coadjunt bishop (if there is one), takes the temporary vacancy:

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.