How do I switch back to Catalina from Monterey?

Please see: Use your iMac as a display with target display mode - Apple Support

Per the link above:

• Your target iMac (monitor) must be using macOS High Sierra 10.13.6 or earlier. 

• In all cases the source Mac that you're connecting FROM must have been introduced in 2019 or earlier and have macOS 10.15 Catalina or earlier installed.

Downgrading your Mac from Monterey back to Catalina will require that you erase the drive and reinstall Catalina.

I strongly recommend that before you wipe that drive you either:

1) make sure that your Mac will boot into recovery properly and display the "Reinstall macOS Catalina" app on the macOS Utilities window: Use macOS Recovery on an Intel-based Mac - Apple Support and/or

2) create a bootable Catalina installer on a USB flash drive and test to make sure that you can boot from that if need be. Create a bootable installer - Apple Support

Also, back up your Mac before you do anything else.

You may want to consider a VNC alternative.

/System/Library/CoreServices/Applications has a Screen Sharing application which allows you to connect to a remote Mac that you can see on your network, even one connected via Thunderbolt. The host Mac you will be viewing on your Screen Sharing Mac just needs Apple menu -> System Preferences -> Screen Sharing turned on.

Screen sharing allows drag and drop of files into and out of the screen sharing window to move files between Macs.

How do I downgrade my Mac OS X operating … - Apple Community