I spent a year building patio covers for a living. YOur price for a 12×20 should be between $3000 and $10K depending on a lot of factors, such as if you use prefab vinyl or wood, and how many decorative enhancements like routering edges and decorative corbels you want. Also, you can save money by going with thinner wood, like 4×4 posts vs more sturdy 6×6 posts, and 1×6 joists vs 2×8’s. YOu’ll have a thinner, toothpick looking result, but you’ll save money.

I disagree that using lattice, or those parallel slats you see on a typical SoCal patio cover, is useless. It blocks out a large percentage of sun and reduces the temp of your patio by a great deal. Full roofs block 100% and lower temps a little more, but lattice still works, and looks nice.

Get three quotes… you should see a range between $3K and $10K.