First of all I think you are confused. The engine oil is not cooled directly by a cooler. The engine its self is cooled by a cooling system. Which uses coolant and a radiator.
Coolant travels through the radiator and is cooled by a combination of factors. There are air dams or cowls that gather moving air and direct it through the fins on the radiator. ther is a fan that is temperature sensitive that comes on and blows air across the fins. Most always, there are transmission coolers built into the radiator, but kept separate from the coolant to prevent the fluid from getting too hot and damaging the transmission. On some radiators there are oil coolers built into the radiator to help cool the oil much like the transmission coolers serve.