中卒で英語は分からんと思うがクーラーは英語でcoolerと書く。

https://www.ldoceonline.com/dictionary/cooler

cool‧er
1. Image of coolera container in which something, especially drinks, is cooled or kept cold
Mike went to fetch a bottle of wine from the cooler.
2. American English a coolbox
3. American English a machine that provides air conditioning
4. → the cooler

3番目の a machine that provides air conditioning
→空調を供給する機械＝エアコン

だからクーラーというのは間違いではないのだ