I have only been able to Sync to Mobile from a single computer.
My other computer indicates that Sync is on but it isn't actually writing anything to the Cloud.
Many people have both a Desktop and a laptop and would probably welcome being able to use both of them interchangeably to Sync Collections to LrM
Only one catalogue can be synced.
So if your catalogue is on an external drive, you can sync from different computers.I've done this with a catalogue on an external drive attached to either my Mac or PC. I'm not confident this is 100% reliable as a supplementary file called Sync.lrdata is stored locally on each machine, and Lightroom seems to use this Sync.lrdata to decide whether to sync images.
I did try moving a copy of my catalogue from one machine to the other but LrM would have none of that trick and refused to accept the imported one!
I will experiment with the catalogue on an external drive but it is not just the Catalogue but also the Smart Previews and another file called Journal which make up the set.
It's not a matter of whether it's on an external or internal drive. At different times I have had the catalogue and its smart previews on the external drive, or on the other computer's internal drive, or on my main computer. Focus on why it "refused to accept the imported one". Was there an error message? For your info, the journal file is purely a temporary file.
Before opening the catalogue on another computer, I delete the Sync.lrdata file as I believe this forces the other computer to rebuild its sync "action list" from scratch. On Mac it's in Users/User/Library/Caches/Adobe/Lightroom, on PC User/AppDate/Local/Adobe/Lightroom/Caches/ . I still don't completely trust it though.
Thank you so much for that invaluable information: the Sync.lrdata file in that location was (and remains!) the source of the problem if one attempts to use Lr Mobile from two different computers that are connected to the same Adobe ID.
When Lr M stops writing Sync files, I quit out of Lr and immediately trash that Cache folder.
I then re-launch Lr and Syncing runs again . . but only for a while until that Cache chokes again.
There may be a capacity-check on that cache? Mine choked at 500 MB and i am now watching it to see if it does the same thing again.