Thunderbolt port late 2013 Imac

Before you ship that mini back, consider that it really is a great Mac, and it will connect to just about any monitor (or TV even) that has an HDMI input. Apple is not especially proprietary regarding monitors these days, and you have options to connect to one. LG, Acer, Dell, ViewSonic and Samsung are just a few name brands that come to mind, and there are lots of off-brand monitors with this input. Prices range from under $100 to several thousand $s. And I know for a fact that Amazon can deliver overnight or sooner! 😃

1) HDMI - pretty standard for many monitors, and the mini supports a 4K monitor at 60Hz. Many monitors do that. This would be my first choice for an inexpensive monitor.

2) Thunderbolt4 - available on many monitors, especially those that are a little more expensive.

3) DisplayPort - a little harder to find in typical monitors, I think, but an option with the appropriate Thunderbolt4 to DisplayPort adapter/cable. This was kind of Apple proprietary at its best.

Here's a link to the specs for the newest Mac minis:

Mac mini - Technical Specifications - Apple

If you are trying to use the iMac as a monitor for the Mac mini, then you are out of luck. That setup is not supported by Apple. The iMac cannot be used as a monitor for any of the newest Macs and Target Display mode is no longer a feature of macOS, having been deprecated several years ago.

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.

However, you might consider a third-party solution, for example: Luna Display or Duet Display.

If you are trying to connect the two computer to transfer data, I would suggest you use a Time Machine backup drive to do that. Create the backup on the old Mac, and then use Migration Assistant on the new Mac to move the data onto it.

Back up your Mac with Time Machine - Apple Support

Restore your Mac from a backup - Apple Support

You're welcome.

Sorry to be the bearer of bad news.

Better news is that there are any number of inexpensive monitors that will have a very nice display for the mini, don't cost an arm and leg, and will be much easier to connect and maintain than that old iMac would have been. 🙂

You're welcome.

Best of luck! 🙂👍🏽