using iMac 21.5 2011 for display with Mac mini m4

You don't, you need to purchase an external display for your MM. iMacs are computers they are NOT external displays for other devices. External displays can be purchased for as little as <$100 or as much as $1500. If you budget about $300 you can get a very nice monitor. Please don't ask which one we suggest though as there are hundreds to select from. What you need to determine is:

• Monitor size
• Monitor resolution (don't forget to cross check against the MM's specifications!!!!!!)
• Budget amount

Then you will at least have a base point on what to look for.

The 21.5" iMac supports Target Display Mode using Thunderbolt input. Unfortunately, Apple added restrictions on the video source. It has to be a Mac released in 2019 or earlier, running Catalina or earlier. The M4 Mac mini is too new on both counts, and does not support using any iMac as a Target Display.

Read the bad news here: Use your iMac as a display with target display mode - Apple Support

So spending $80+ to make a Thunderbolt connection between the Macs won't let you use the old one as a display, and even if it did, it might not be very economically attractive given the size and resolution of that old iMac screen, and the prices of standalone displays today.

The M4 Mac mini does not support using your old 2011 or any other iMac in a cabled Display setup.

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

If you have or get an M4 Mac mini, then you need to get and use an external display.

see > Connect a display to Mac mini - Apple Support

Without a separate monitor, you can't. If you have one, set up Virtual Network Computing; the cabling or wireless connection used is the same as the one needed to connect them to a router.

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