The Mouth of Sauron

[Gwaihir the Windlord]

In other words, it is possible that the Mouth of Sauron = Gothmog.

If this is true, then the poor bugger would have been both Lieutenant of Barad-dur and of Minas Morgul, at the same time. I would not have thought that this would have been the same person. As Lieutenant of Minas Morgul, Gothmog would primarily have been an overseer of the war machine there. The Mouth, as Lieutant of Barad-dur, was second-in-command of Mordor. As Legolas rightly says, the Nazgul were special agents as well as wielders of great force that reported directly to Sauron; his personal Secret Service and a driving force of his armies. They were not administrators.