iMac as a monitor

While your Mac computers do not work with Target Display mode, you could look into a third-party option, such as Luna Display. However, your iMac cannot be used as the sole display for a Mac mini, but, you can use the iMac as a second display using Luna Display.

- Jack

+1 As mentioned, you need an external display to setup and use as the primary display.

To connect a display or two up to a 2023 Mac mini, see > Connect a display to Mac mini - Apple Support

Buy yourself a 4K LG 32-inch display for your new Mac mini and rejoice in the extra screen real estate that is nicer than the 2013 iMac can provide. Once you experience 32 inches, you will squint in front of an iMac.

Here is an example of screen sharing my 2020 Core i7 iMac (Monterey 12.6.8) to my 2023 M2 Mac mini Pro (Ventura 13.5). The iMac is running Parallels Desktop Pro 18.3 Virtual Machine with Windows 11 Pro as a full screen guest. Both the iMac and the mini are connected by Wi-Fi 6. This is an LG 32-in UltraFine 4K display.

the good news is a normal monitor will be using much much much less power than using another computer as display