Use Late 2013 iMac as monitor
I have an iMac 27" (Late 2013) running macOS Catalina vs 10.15.7.
Can I buy a current model Mac mini M2 and use the 27" iMac as a
(dumb) monitor?
Mac mini (M2, 2023)
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Nope. Target Display Mode used Macs that met certain very specific requirements, and yours are not among them.
Use your iMac as a display with target display mode - Apple Support