Using Time Machine to backup a NAS that is connected to my Mac mini

No, you cannot directly use Time Machine to back up a Network-Attached Storage (NAS) device onto a local external SSD, because Time Machine is designed to back up a Mac to a storage volume, not a NAS to another drive. An attached external drive can be backed up using Time Machine, but not an NAS (even mounted as a share).

To back up the actual contents of your NAS onto an external SSD, you should use your NAS's native backup application, Synology's Hyper Backup.

https://www.synology.com/en-us/dsm/feature/hyper_backup

The important distinction is that Time Machine backs up the Mac itself, not arbitrary network shares that happen to be mounted in Finder, so simply mounting the Synology volume on your Mac mini will not make its contents part of the Time Machine backup.

In your setup, the clean solution is to let the DS118 handle its own backup with Hyper Backup and target the 2TB USB drive attached to the Mac only if you can expose that drive to the NAS as a network destination; otherwise, attach the USB drive directly to the Synology and run Hyper Backup locally. Since your data is already mirrored to another NAS, I’d also avoid relying on the Mac mini as the middleman because it adds another device and failure point.

If your goal is specifically to have Time Machine manage everything, you’d need to first copy/sync the NAS data into storage that macOS treats as local Mac data, but that’s generally a much less sensible architecture for a 2TB genealogy archive.

Thanks everyone. That was helpful.

Bob