Mirroring your phone screen has always been a tough issue to nail for developers and hardware manufacturers alike, there are several competing standards out there like Miracast, WiFi Direct, the Chromecast, and many many others. Additionally, there’s also no easy way to make these standards work with each other, since they all use different technology to accomplish the same goals.
With more smart projectors coming out, running full operating systems like Android, this is becoming easier to do. There are apps out there like MirrorOP, which when installed on two devices, will allow them to mirror their devices to each other. This is pretty easy to setup in practice, it involves installing the two apps on the devices and connecting them via WiFi. The video I posted below demonstrates this setup using a Samsung Galaxy S4 and the AAXA Technologies LED Android projector.