What does the padlock icon next to a name mean in staff management

When the Controller icon appears, it simply means you can affect the scene (usually by simply moving the camera).

In my experience, its a visual / aesthetic option only, and you should be perfectly fine to ignore your controller during these moments.

Staff Morale effects your soldiers abilities. When they are in good morale, they will perform better than their skills, indicated by a yellow boost to their bars:

The primary factor for Morale is GMP balance. Keeping a surplus is the best way to ensure that Morale is not negatively effected. You can also visit the soldiers individually at Mother Base to raise their morale.

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