I looked at the Adidas web site, and it does state that the hat provides UPF protection, but it doesn't say how much. The OR Swift provides 50+, even with the mesh. Many hats that have mesh provide very low protection through the mesh, even if the rest of the hat is great.
I have the Sunday Afternoons Sun Guide Cap, which is very similar to the OR Sun Runner, but with a longer brim in the front. It works, but it can still get warm under the cape. It's not super hot, but it would certainly be better to have open ventilation there, if I can still get enough sun protection from Da Brim. I will find out soon whether my new idea will provide that protection or not, because my Da Brim should arrive any day.