Detail the specific functions and operational scenarios where an Auxiliary Power Unit (APU) is critical for a locomotive, differentiating its role from the main prime mover, particularly concerning hotel power, climate control, and main engine cold start support.

An Auxiliary Power Unit (APU) in a locomotive is a small, independent internal combustion engine coupled with a generator, entirely separate from the locomotive's main prime mover. The main prime mover is the large diesel engine designed to generate massive power for the locomotive's traction motors, which propel the train. The APU's criticality stems from its ability to provide essential power for auxiliary systems without requiring the operation of the large, fuel-intensive main engine, thereby optimizing fuel efficiency, reducing wear, and enabling crucial functions in specific scenarios. Regarding hotel power, the APU is critical for supplying electrical power to all non-traction systems onboard the locomotive when the main prime mover is shut down or idling. Hotel power encompasses crucial functions such as lighting within the cab and car body, communication systems like radios and data links, onboard computers, safety equipment like event reco....