Eminence was a form of address for some religious figures.

The term could be used to refer to the Kai of Bajor, including Winn Adami between 2371 and 2375. (DS9: "Life Support", "Shakaar", "In the Cards", "'Til Death Do Us Part", "Strange Bedfellows", "The Changing Face of Evil", "When It Rains...")

After Winn and Gul Dukat had entered the Fire Caves together in late 2375, Winn instructed Dukat to address her as "eminence" and no longer as "Adami", telling him he had lost that privilege. (DS9: "What You Leave Behind")

In reality, eminence is the traditional form of address for cardinals. Unlike Winn, Kai Opaka was never addressed as eminence, just as Kai or Opaka.
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