target display mode iMac late 2013 macOS Catalina

None of the new Mac’s can use an iMac in Target Display Mode.

carefully read > Use your iMac as a display with target display mode - Apple Support

If have or buy a Mac Studio or a Mac mini, then you will need an external display to setup and use as the main display.

see > Connect a display to Mac Studio - Apple Support

Then you can repurpose or sell the 2013 iMac.

sorry, gori666, but Target Display Mode has been dead a long time.... It might technically work over thunderbolt, but there are certain requirements for both Macs, and your Mac Studio, being an M1/M2 Mac definitely doesn't support target display mode. You can get/buy "Luna Display/Duet Display" which will allow you to use your late 2013 imac as a monitor for your Mac studio...that's the only thing I know which will, so yes, with "Luna Display/Duet Display" it is possible

John B