Does the About the Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter support an iMac 2013?

It may not work for a 2020 Mac to use the old iMac as a display.

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

"You can connect any of these older iMac models to another Mac from 2019 or earlier"

There have been reports of success with a 2020 iMac using a 2009 iMac as a display, but that uses Mini DisplayPort, not Thunderbolt. For your 20132 iMac you would need a Thunderbolt 3 adapter

https://www.apple.com/shop/product/MMEL2AM/A/thunderbolt-3-usb-c-to-thunderbolt-2-adapter

and a Thunderbolt cable.

Apple Thunderbolt Cable (0.5 m) - White - Apple

Let us know if it works.

No reason it wouldn't work, unless there is a hardware problem. 

Depends what you are using it to connect to.

Someone in another thread said they tried to connect a new

2020 iMac to a 2013 27" and could not get it to work.