The correct answer is A) around the cooler. The three-way thermostatic control valve regulates the diesel engine cooling water temperature by passing a portion of the water around the cooler. This allows the valve to control the amount of cooling water that flows through the cooler, thereby regulating the engine's operating temperature. The other options are incorrect because: B) Around the engine would not regulate the temperature, as the water would still be circulating through the engine. C) Overboard would completely bypass the cooling system, which would not regulate the temperature. D) To the expansion tank would not directly regulate the temperature, as the expansion tank is part of the overall cooling system.