Moving photo library

You can move the Photos Library to an external drive and make it your system photos library and sync it with iCloud, if the external drive is compatible with Photos. It has to be prepared as described here: Move your Photos library to save space on your Mac - Apple Support

Basically,

• the file system format needs to be MacOS Extended (Journaled) or APFS,
• the drive needs to have a wired connection - you cannot store the Photos Library on a network share
• the volume with the library must not have been used for Time Machine backups or have other backup software installed
• the library must not be in a synced folder like the DropBox, iCloud Drive, or Google Drive.

If all your photos are already in iCloud Photos, it may be more economic to create a new , empty library on your external volume and make this new library your System Photos Library and enable iCloud Photos for it. Then your library will simply download from iCloud to the new library and and you can continue where you left off. Starting over with an empty library will save the lengthy, new upload to iCloud, if you move the complete library to the external volume. If you move a library, that is syncing with iCloud, you have to make it again your iCloud Library, and that will cause a new upload to iCloud to compare it to the existing library. This can take a very long time, if the library is large.