There's an Office of Sustainability at the University of Wisconsin, complete with a video production budget. "The University of Wisconsin Office of Sustainability posted a video to educate UW students on how to properly recycle their coffee cups." Only the cup is "food waste." The lid and the wrapper are recyclable. But field-stripping a coffee cup takes more time than a lot of harried students, staff, or faculty have, and the cups along with the impedimenta all get tossed. And somewhere a deanlet has her undies in a bundle. Thus the video.
But the presence of a wrapper poses a question ... are the cups themselves flimsier and more prone to transmit heat now, making the wrapper obligatory, rather than something prompted by a stupid lawsuit a few years ago? My impression ... or perhaps it's just advancing age ... is that the paper cups are hotter to the touch now than they used to be.