Moved to the Lightroom forum
LR Mobile is for connecting, syncing, One Catalog on One computer to Multiple Mobile devices. It is not for syncing multiple computers to one catalog.
So if I read your question right the answer is No it can't do that.
The catalog, if there is a catalog, on a mobile device is completely different from the LR catalog on a computer. The sync happens from Collections you create on both mobile device and computer not the actual catalog file.
My question is, do the mobile counterparts of this software allow you work between the two programs and store back in Lightroom ? The catalogues are a must have and easy to share right from my phone without having to go into my MacBook. thanks in advance !
All work you do in Lightroom Mobile will sync back to your Lightroom catalogue.
When you work in other mobile apps, like Photoshop Fix or Photoshop Mix, there is generally an option to save back to Lightroom. The edited file then appears in Lightroom Mobile and is synced back to your Lightroom catalogue.