When the factory compresses the air to make the can of compressed air, it’s warm from the compression (if you released it all right then) it would drop back to the original pre-compression temperature. Hopefully that makes sense.
Letting it cool first, doesn’t change that second temperature drop, it just means that the gas drops from a lower starting point so it gets very cold.
Those two principles are the trick that makes refrigeration and air conditioning work. We do the compression and decompression over and over again with the same gas inside and outside the thing we want to be cool.
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