In my travels, I have seen many approaches to getting various computer and video signals to a projector. A lot of what I have seen involves running all the different signals to the projector and using the remote to switch between the inputs. I’ve also seen cheap consumer-grade AV switchers where the computer signal is down-converted to a composite or S-Video signal, then run through the AV switcher, then up to the projector.
I’m here to tell you there is a better, cleaner, and more professional way of accomplishing this.