You know the difference, right? Auxiliaries have no right of succession; co-adjutors generally do. Christus Dominus 25-26 lays out the need for a bishop who needs help with a demanding job and stresses the need for unity among multiple bishops in a single diocese.
A coadjutor bishop, appointed with the right of succession, must always be named vicar general by the diocesan bishop. In particular cases the competent authority can grant him even more extensive faculties.
The celebrated case of “even more extensive faculties” was of Donald Wuerl being appointed in Seattle under Raymond Hunthausen.