When you've decided on a satisfactory set of internal settings for the projector then you can then profile it. The resulting profile will ‘drive’ the output from your computer to the projector. But then, of course, you must ensure that the internal settings of the projector do not get changed or this would invalidate what the profile is trying to do.
Thanks for the reply. I have the projector set up for sRGB with Computer as the input. I agree with your procedure. What I did was simply lower the brightness setting on the projector by 6 points. Reducing contrast by even 1 point made the very darkest tones muddy, so the only change was to the brightness. I then profiled with the ColorMunki.
Another person said that I should profile first and then reduce the brightness on the projector. His reasoning was as follows: If I reduce the brightness first, the ColorMunki would then do a profile that would try to compensate for the lowered brightness, and I would end up with a too-bright profile again. That point confuses me, and I still believe the correct way would be to adjust the brightness of the projector and then profile.