Upgraded Lightroom and Catalogue does not see my external Hardrive on the desktop.

Since the external hard drive appears on your desktop and in Finder, it seems to be properly connected and recognized by the operating system. However, Lightroom is not displaying it as an option under the "source" dropdown menu.

One thing you might try is to restart both Lightroom and your MacBook Pro. Sometimes, simply restarting the software or the computer can resolve connectivity issues with external drives.

If restarting doesn't resolve the issue, you could also try disconnecting and reconnecting the external hard drive to your MacBook Pro. This can sometimes trigger the system to recognize the drive again.

Additionally, you may want to check if Lightroom has the necessary permissions to access external drives. You can do this by going to System Preferences > Security & Privacy > Privacy tab, then selecting "Files and Folders" from the left sidebar. Make sure Lightroom is listed and has permissions to access external drives.

If none of these steps resolve the issue, it's possible that there may be compatibility issues between Lightroom and macOS Monterey. In this case, you might consider reaching out to Adobe Support for further assistance or checking their support forums to see if others have encountered similar issues and found solutions.

A few things to try ...

Give Lightroom Full Disk Access. Go to System Preferences > Security & Privacy > Privacy > Full Disk Access. Add Lightroom to the list.

After doing Full Disk Access, in Lightroom do a plain File > Open Catalog and see if you can navigate to the external drive and select your catalog.

ps. I am using LR Classic v13.1 and I don't see any prompt for "Source" when I try to open a catalog. Where/when are you seeing this prompt?

Finally, have you tried just double-clicking the v10 .lrcat to see if it opens in Lightroom 13? Note: because LR13 will upgrade the LR10 catalog, it would be a good idea to first make a backup copy of the LR10 catalog.

Keep it in mind in the future. For most users Full Disk Access is still a somewhat obscure security feature in macOS. Prior to its advent all applications had complete access to any & all drives & folders on a Mac. Now, the default security is applications only have access to certain limited locations such as Desktop or Documents. Even so, the user may be asked for permission for the app to access even those locations. That's ok for many applications but not all. Lightroom, Photoshop and similar "more powerful" apps often need to be given Full Disk Access.

What version of macOS did you upgrade from, and what version of Monterey do you have?

What version of Lightroom are you using?

But what version of macOS did you upgrade from?

And what version of Lightroom did you have before & after upgrade?