Logic Pro Storing two Logic Pro sound libraries on one external hard drive??

Hello fellow LUG'ers

I've got two Macbook Air laptops running Logic Pro 10.7.9 (one for my teaching job, and one for my professional work), and I'm wondering if I can store both instances of the application's sound libraries on the same external hard drive? My external HD has lots of space available, so it makes sense to use it.

However, when I open up Logic Pro on my pro-machine and go to the Relocate Library function, the external HD is not coming up as an option, and it appears that the issue is that the HD is already storing the sound library for my teaching machine. I've attached a screen to show the message I get ('Sound Library Files exist here).
Screen Shot 2023-09-10 at 11.46.59 am.png

Does this mean that I need to buy/use another external HD if I want to relocate the sound library of my pro-installation of Logic Pro?

Many thanks in advance for any help!
Andrew Harrison