How do I move a photos system library from a MacBook Air to a desktop Mac?

If your Photos library on the MacBook Air is an iCloud Photo Library, you could simply create a new, empty Photos Library on the desktop Mac and enable it for iCloud. Then wait for the library to download to the Desktop Machine.

There would be one draw-back. On Sierra the faces do not sync to iCloud like they do on High Sierra. So the named faces would not sync to the Desktop Mac. Also the projects would be missing. Use Photos and iCloud Photo Library on multiple Mac computers - Apple Support

Same as you would any other item. Link the two machines and drag and drop, or, drag it to an external HD and then hook that to the other machine and drag it over.

Be aware that if there is an existing Library on the target machine then it risks overwriting it, so be careful about that. Rename one or other.