Back in 2011 I wrote a long technical post about mirroring an Android phone to a laptop using the Android SDK, Java, USB debugging mode, and a screen capture tool called Ashot. The use case was the same one I am about to describe. The procedure is unrecognizable now.
The reasons a lawyer might want to do this have not changed. You may want to show a phone screen to opposing counsel during a negotiation, to a jury at trial, or to an audience during a CLE presentation. Sometimes you want to capture video or screenshots of phone activity for use in a deposition exhibit or a slide deck. None of that requires the SDK route anymore.
For iPhone
If you have a Mac, the cleanest method is QuickTime Player. Plug the phone into the Mac with a cable, open QuickTime, choose File, then New Movie Recording, and select the iPhone as the camera source. The screen mirrors to the Mac in real time, and you can record it.
If you want to skip the laptop entirely, use AirPlay to an Apple TV connected to the screen, or use a USB-C to HDMI adapter (Lightning to HDMI for older iPhones) and plug straight into the projector.
For Android
A USB-C to HDMI adapter plugged into the projector is the simplest option for most modern Android phones. Check that your phone model supports video output over USB-C, because not all do.
If you want to mirror to a laptop, scrcpy is free, open source, and works over USB. Much easier than what 2011 required.
If a Chromecast is available at the venue, you can cast directly from Android.
Either Platform
If you are already on a Zoom or Teams call, the simplest option is often to share your phone screen through the video platform. Both apps support iPhone and Android mirroring.
A Few Practical Tips
Test the setup at the venue before the event. Courtroom AV is unpredictable. Bring backup screenshots or video on a thumb drive in case the live mirroring fails.
Turn on Do Not Disturb before you start. The last thing you want is a personal text or a notification popping up in front of a jury.
Consider whether you really need a live screen at all. A screenshot taken in advance, blown up on a slide, often works as well and carries none of the risk.