I placed the youtube embedded code on a website that is both desktop & mobile friendly. The desktop version accepts the youtube video in it's design nicely. The mobile version on the other hand looks awkward with the youtube video being wider then the whole mobile design itself.
---- <----mobile design width
_______ <----youtube video width wider then the mobile design
Never seen that on any of the sites I've seen embedded videos on.
Plus, you can full-screen in mobile anyway which will fit the video to the entire display.