[Adoby]

You need to setup some form of sync between your separate installs. That is not trivial to do, unfortunately.

Some here use cloud services, others use some local sync software.

The problem is that calibre is a strictly single user program, and placing the calibre library on a shared networked filesystem is known to cause problems.

Make sure you have a good backup system and fresh backups before you attempt to sync between computers. Otherwise you may loose data and books.

Restoring the backup from one computer to the other may be enough to achieve what you want, especially if you not only backup the calibre library but also the configuration folder, but it won't help you to keep the separate libraries synced.

Another option that might be enough is to only have calibre installed on one computer and access that install remotely, either using the browser and the content server or using remote desktop software.

Yet another method could be to use a portable install of calibre on a thumb drive, and when needed move that thumb drive between computers.

You can find a lot of posts here that discuss this. Look below for some similar threads.