Need to mirror MacBook Air m2 to IMac 21.5 late 2013
Running Sonoma 14.1.1 on MacBook Air. Running high sierra 10.13.6 on iMac 2013.
Imac 2013 does not show up on MacBook Air dropdown as a mirroring option. Please help. Thank you.
MacBook Air
It sounds like you're trying to use the 2013 iMac as a display for the M2 MacBook Air. That won't work.
Based on your reference to a dropdown menu, it appears that you were trying to use AirPlay. Macs have been able to use AirPlay to send video to Apple TV set-top boxes for a very long time. The ability to AirPlay to a Mac (AirPlay to Mac) is much more recent. For an iMac, you'd need an iMac introduced in 2019 or later, running macOS Monterey or later.
Aside: I'm guessing that you might be able to set up your M2 MacBook Air as an AirPlay Receiver for AirPlay to Mac, and send video from your 2013 iMac to your M2 MacBook Air. I'm also guessing that the 2013 iMac will be using an older version of AirPlay and will be limited to an AirPlay resolution of at most 1920x1080 pixels.
System requirements for Continuity features on Apple devices - Apple Support
Set up your Mac to be an AirPlay Receiver - Apple Support
Target Display Mode won't help you either. Your 2013 iMac supports Target Display Mode using Thunderbolt input (plain Mini DisplayPort input won't work). But Apple added restrictions on the video source so that the source has to be another Mac, released in 2019 or earlier, running macOS Catalina or earlier. Your M2 MacBook Air is too new to use ANY iMac as a Target Display.
Use your iMac as a display with target display mode - Apple Support
Sounds like it may be time for you to go shopping for a standalone external monitor.