Yes, it should work. From my own experience, I can only confirm that you can use it to connect iPads with USB-C to the TV. This will work in exactly the same way with the new iPhone 15.

PS: How well the size/resolution of the TV is utilised as a 2nd display depends heavily on the iOS apps. Certain apps such as PlayerXtreme automatically recognise when a 2nd display is connected and only play the video/film/series on the 2nd display, while only the video control remains on the iPad/iPhone. Unfortunately, other apps such as Youtube are not yet able to do this and you can then only mirror the entire iPad/iPhone screen on the 2nd display, which means that the resolution is not perfectly adapted to the 2nd display.

I received the adapter today. The picture transfer from the iPhone to the TV via HDMI works perfectly.

It works..... even a keyboard or mouse. But watch out for the various applications. Some do and some don't. Especially streaming applications