Canon 403. §1 When the pastoral needs of the diocese require it, one or more auxiliary
Bishops are to be appointed at the request of the diocesan Bishop. An auxiliary
Bishop does not have the right of succession.

§2 In more serious circumstances, even of a personal nature, the diocesan Bishop may be given an auxiliary Bishop with special faculties.

§3 If the Holy See considers it more opportune, it can ex officio appoint a coadjutor
Bishop, who also has special faculties. A coadjutor Bishop has the right of succession.

§1. Cum pastorales dioecesis necessitates id suadeant, unus vel plures Episcopi auxiliares, petente Episcopo dioecesano, constituantur; Episcopus auxiliaris iure successionis non gaudet.

§2. Gravioribus in adiunctis, etiam indolis personalis, Episcopo dioecesano dari potest Episcopus auxiliaris specialibus instructus facultatibus.

§3. Sancta Sedes, si magis opportunum id ipsi videatur, ex officio constituere potest Episcopum coadiutorem, qui et ipse specialibus instruitur facultatibus; Episcopus coadiutor iure successionis gaudet.

Canon 404. §1 The coadjutor Bishop takes possession of his office when, either personally or by proxy, he shows the apostolic letters of appointment to the diocesan
Bishop and the college of consultors, in the presence of the chancellor of the curia, who makes a record of the fact.

§2 An auxiliary Bishop takes possession of his office when he shows his apostolic letters of appointment to the diocesan Bishop, in the presence of the chancellor of the curia, who makes a record of the fact.

§3 If the diocesan Bishop is wholly impeded, it is sufficient that either the coadjutor
Bishop or the auxiliary Bishop show their apostolic letters of appointment to the college of consultors, in the presence of the chancellor of the curia.

§1. Episcopus coadiutor officii sui possessionem capit, cum litteras apostolicas nominationis, per se vel per procuratorem, ostenderit Episcopo dioecesano atque collegio consultorum, praesente curiae cancellario, qui rem in acta referat.

§2. Episcopus auxiliaris officii sui possessionem capit, cum litteras apostolicas nominationis ostenderit Episcopo dioecesano, praesente curiae cancellario, qui rem in acta referat.

§3. Quod si Episcopus dioecesanus plene sit impeditus, sufficit ut tum Episcopus coadiutor, tum Episcopus auxiliaris litteras apostolicas nominationis ostendant collegio consultorum, praesente curiae cancellario.

Canon 405. §1 The coadjutor Bishop and the auxiliary Bishop have the obligations and the rights which are determined by the provisions of the following canons and defined in their letters of appointment.

§2 The coadjutor Bishop, or the auxiliary Bishop mentioned in can. 403 §2, assists the diocesan Bishop in the entire governance of the diocese, and takes his place when he is absent or impeded.

§1. Episcopus coadiutor, itemque Episcopus auxiliaris, obligationes et iura habent quae determinantur praescriptis canonum qui sequuntur, atque in litteris suae nominationis definiuntur.

§2. Episcopus coadiutor et Episcopus auxiliaris, de quo in can. 403, §2, Episcopo dioecesano in universo dioecesis regimine adstant atque eiusdem absentis vel impediti vices supplent.

Canon 406. §1 The coadjutor Bishop, and likewise the auxiliary Bishop mentioned in can. 403 §2, is to be appointed a Vicar general by the diocesan Bishop. The diocesan
Bishop is to entrust to him, in preference to others, those things which by law require a special mandate.

§2 Unless the apostolic letters provide otherwise, and without prejudice to the provision of §1, the diocesan Bishop is to appoint his auxiliary or auxiliaries as Vicar general or at least episcopal Vicar, in dependence solely on his authority, or on that of the coadjutor Bishop or of the auxiliary Bishop mentioned in can. 403 §2.

§1. Episcopus coadiutor, itemque Episcopus auxiliaris, de quo in can. 403, §2, ab Episcopo dioecesano Vicarius generalis constituatur; insuper ipsi prae ceteris Episcopus dioecesanus committat quae ex iure mandatum speciale requirant.

§2. Nisi in litteris apostolicis aliud provisum fuerit et firmo praescripto §1, Episcopus dioecesanus auxiliarem vel auxiliares suos constituat Vicarios generales vel saltem Vicarios episcopales, ab auctoritate sua, aut Episcopi coadiutoris vel Episcopi auxiliaris de quo in can. 403, §2, dumtaxat dependentes.

Canon 407. §1 For the greatest present and future good of the diocese, the diocesan
Bishop, the coadjutor and the auxiliary Bishop mentioned in can. 403 §2, are to consult with each other on matters of greater importance.

§2 In assessing matters of greater importance, particularly those of a pastoral nature, the diocesan Bishop is to consult the auxiliary Bishop before all others.

§3 The coadjutor Bishop and the auxiliary Bishop, since they are called to share in the cares of the diocesan Bishop, should so exercise their office that they act and think in accord with him.

§1. Ut quam maxime praesenti et futuro dioecesis bono faveatur, Episcopus dioecesanus, coadiutor atque Episcopus auxiliaris de quo in can. 403, §2, in rebus maioris momenti sese invicem consulant.

§2. Episcopus dioecesanus in perpendendis causis maioris momenti, praesertim indolis pastoralis, Episcopos auxiliares prae ceteris consulere velit.

§3. Episcopus coadiutor et Episcopus auxiliaris, quippe qui in partem sollicitudinis Episcopi dioecesani vocati sint, munia sua ita exerceant, ut concordi cum ipso opera et animo procedant.

Canon 408. §1 As often as they are requested to do so by the diocesan Bishop, a coadjutor Bishop and an auxiliary Bishop who are not lawfully impeded, are obliged to perform those pontifical and other functions to which the diocesan Bishop is bound.

§2 Those episcopal rights and functions which the coadjutor can exercise are not habitually to be entrusted to another by the diocesan Bishop.

§1. Episcopus coadiutor et Episcopus auxiliaris, iusto impedimento non detenti, obligantur ut, quoties Episcopus dioecesanus id requirat, pontificalia et alias functiones obeant, ad quas Episcopus dioecesanus tenetur.

§2. Quae episcopalia iura et functiones Episcopus coadiutor aut auxiliaris potest exercere, Episcopus dioecesanus habitualiter alii ne committat.

Canon 409. §1 When the episcopal see falls vacant, the coadjutor immediately becomes the Bishop of the diocese for which he was appointed, provided he has lawfully taken possession.

§2 Unless the competent authority has provided otherwise, when the episcopal see is vacant and until the new Bishop takes possession of the see, the auxiliary Bishop retains all and only those powers and faculties which he had as Vicar general or as episcopal Vicar when the see was occupied. If he is not appointed to the office of diocesan Administrator, he is to exercise this same power of his, conferred by the law, under the authority of the diocesan Administrator, who governs the diocese.

§1. Vacante sede episcopali, Episcopus coadiutor statim fit Episcopus dioecesis pro qua fuerat constitutus, dummodo possessionem legitime ceperit.

§2. Vacante sede episcopali, nisi aliud a competenti auctoritate statutum fuerit, Episcopus auxiliaris, donec novus Episcopus possessionem sedis ceperit, omnes et solas servat potestates et facultates quibus sede plena, tamquam Vicarius generalis vel tamquam Vicarius episcopalis, gaudebat; quod si ad munus Administratoris dioecesani, qui regimini dioecesis praeest.

Canon 410. The coadjutor Bishop and the auxiliary Bishop are bound, like the diocesan
Bishop, to reside in the diocese. Other than for the fulfilment of some duty outside the diocese, or for holidays, which are not to be longer than one month, they may not be away from the diocese except for a brief period.

Episcopus coadiutor et Episcopus auxiliaris obligatione tenentur, sicut et ipse Episcopus dioecesanus, residendi in dioecesi; a qua, praeterquam ratione alicuius officii extra dioecesim implendi aut feriarum causa, quae ultra mensem ne protrahantur, nonnisi ad breve tempus discedant.

Canon 411. The provisions of cann. 401 and 402 §2, concerning resignation from office, apply also to a coadjutor and an auxiliary Bishop.
[NB see m.p. Learn to take your leave, 12.II.2018, art. 5 (cf. can. 401)]

Episcopo coadiutori et auxiliari, ad renuntiationem ab officio quod attinet, applicantur praescripta can. 401 et 402, §2.