One interesting thing I have noticed about my "fit over the end of a Nalgene bottle" cup is that it will not fit over the end of a polycarbonate bottle of the same capacity. Go figure.

As for carrying the homemade "lid" for the cup, I made a little "stove" for the mug from a 12 oz coffee can. That fits (very tightly) inside of the non-insulated carrier I have for the bottle. I used the cut off top from another tin can (the coffee can had one of those foil tear-off tops) as the lid for the cup, it fits inside of the stove, the cup slides inside of the stove, the bottle inside of the stove and cup. Not real fast to remove from the carrier, not the best lid in the world, but at least it is all there in one place...