Why is my Apple TV name not showing up in mirroring ?

Hello Gfunk2022,

Welcome to Apple Support Communities!

If we understand your post correctly, Apple TV is not showing up on the list of devices to mirror your iPhone with. We'd like to help with that.

"You can use AirPlay to:

• Play videos or audio from your iOS or iPadOS device on Apple TV
• Play music, podcasts, and radio stations on multiple AirPlay 2–enabled devices (such as HomePod or other speakers) in rooms throughout your house (see Use Apple TV to play audio throughout your home)
• View the screen of your Mac, iOS device, or iPadOS device on Apple TV

To stream content from a Mac, iOS device, or iPadOS device to Apple TV, both Apple TV and the shared device may need to be connected to the same network. For more information, see the Apple Support article Use AirPlay to stream video or mirror your device’s screen."

More information can be found in this article: Use AirPlay to stream audio and video on Apple TV

Let us know if you have any questions.

Thank you for using Apple Support Communities.

Take care!