I’ve got some CO2 cartridges intended for inflating bicycle tyres; to attach them to the bicycle valve I use a right-angled adapter with a built-in spring valve; when it’s all attached, the sprung part is depressed, releasing the gas and inflating the tyre. The setup, without the wheel, looks a little like this:

When I use it to inflate a tyre, the CO2 cylinder gets very cold – water condenses and then freezes on it. This is expected, and is due to adiabatic cooling:as the gas expands, it extracts heat from around it. The metal adapter also freezes, though. While there is some expansion in the adapter, I’d have expected the friction of the gas to heat it more than it’s being cooled, but it seems to get colder than the cylinder (subjectively – I haven’t measured it.)
Why does the adapter cool so much?