Part of macOS
corespotlightdSpotlight for apps
○ Leave it
Indexes what apps share with Spotlight, such as mail and notes.
What it is
Lets apps add their own items, like emails and messages, to Spotlight search.
Why it might be busy
After a large mailbox or library changes. Usually a few minutes.
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
corespotlightdmanagedcorespotlightd
From Procession’s guide, in the words the app shows.