The biggest responsibility of any oil is to form a layer between metal surfaces of various engine parts, in a transmission or in the differentials. This layer is what provides the lubrication characteristics of any oil. Oil serves as a sealant, filling the microscopic ridges and valleys found in any metal surface, increasing the engine's efficiency. Oil must serve as a cleaning function, carrying away dirt or other debris which damages bearings or other parts which operate in tight tolerances. Debris is removed through the engine oil filter or the transmission filter.
Oil uses detergent additives to combat combustion by-products.
The detergent fights these by-products, inhibiting their buildup as sludge, varnishes, etc. Oil is a tremendous coolant. In the engine, the oil cools the underside of the pistons, valve springs, camshaft, rods, crankshaft and bearings. That could very likely eliminate the need for an oil cooler altogether. Oil is the only coolant in your automatic transmission.