Part of macOS
SoftwareUpdateNotificationManagerUpdate reminders
○ Leave it
Reminds you about macOS updates.
What it is
Shows update notifications and finishes tasks after an update.
If you stop it
macOS starts it again straight away. Leave it.
The verdict
○ Leave it. Part of macOS, or something that starts again by itself. You never need to close these.
In Activity Monitor
SoftwareUpdateNotificationManagerUpdateSettingsToolpostupgraded
From Procession’s guide, in the words the app shows.