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Do i need a oil cooler for automatic transmission for trailer towing? - trailer cooler rack

Hey Guys, are'm not sure you need an oil cooler for automatic transmission .. 3.2ltr turbo diesel in a van .. I am a trailer with a total weight of 750 kg and the load does not exceed 300 kg .. Is it true that the gearbox can suffer serious damage in the event of overheating, too?

no idea what I should do would be great ..

Thanks