Will 4x4 posts carry a patio cover with a 12 foot span?
So 4x4 posts every 7 feet, single 2x6 beams, and 2x6 rafters spanning about 12 feet. Right?
That's all fine except for the beams. They're barely adequate and will probably sag. You don't have to worry about snow loads, but you'll have a wavy roof after a few years.
I'd double them and fasten them together with two screws every 16 inches or so. Don't use a flat 2x4. That does nothing as a board like that is a trampoline.