Is it safe to disconnect external hard drive where macOS was installed?

Hi, yes, that's actually what I meant. I'm running catalina from my external drive. It happened because I used the external drive when installing catalina because I couldn't see the macintosh hd or main drive that time (it wasn't in the option). Even when I installed El Capitan it was running from external drive as it did the same thing with catalina, I couldn't find the macintosh hd or main drive. Do you have any suggestion on where would I start? Thanks so much for your time

After installing Catalina, macintosh hd and macintosh hd data are visible but macintosh hd is not visible in start up disk no matter how many times I tried to restart. So does it mean I had to go all over again restoring to factory settings? I want to use macintosh hd as start up disk.

Hi there, thank you so much for checking in. Actually, it's half fixed. What I meant is my MacBook is running well with Catalina at the moment. However, I couldn't fix my main issue with macintosh hd. I'm still running macOS catalina from external drive, and I have been reading all the related issues in apple support but still can't fix it. I tried restoring it again and checking the issues in disk utility but macintosh hd seems fine. Although I can't use it as start up disk, that's why when I'm trying to restore macOS, it's only showing my external drive as an option. So, I'm quite stuck with my external drive. Do you have any suggestions? Thanks for your time again.