Friction heating is correct my good man. Technically you are increasing TAT (total air temperature).
Descending would not increase your speed, necessarily. Actually, you would decrease Mach #, usually, by descending. Not that that really has anything to do with, well, anything. These thing's don't change pitch angle much at all when climbing or descending, in fact the 777 mostly increases power for a cruise climb (opposite for a descent), and doesn't deviate too much off of 2 1/2 degrees.
Normally one would decrease altitude to increase air temperature, so the two thing's are unrelated other than they have the same effect on the fuel temperature.
Likely there is someone here who can offer a really good technical answer, but that's just the basics of it from my view point.