I don't want to do that, but WHILE building the library I think it's normal to have the code for the library LOCAL in an extra TAB - right, and when you're finished building the library, you want to make it a 'normal library'.
But where to store it so it can be use from both OFFICE and LAB in future projects - that my question.
You have two issues. Making libraries and storing libraries for remote access.
Follow the link from @hammy to make your libraries. If you want to access them from multiple location you can store them in a cloud, github, or any of a 1000 other free options, already meant specifically for this, to choose from. Be aware though that using a cloud drive may require you to use a portable version of the IDE.
Look into github, it's probably your easiest option and is one of the most used by the programming community. You can access it from any location and commit changes from any location. Gone are the days of USB flash drives . . . and misplacing them