The computer I am using is through the government agency where I work on site. Do I just leave my equipment, put my code somewhere, and go tell my supervisor for the contracting company with a written letter?
At some point in you're life you're likely to get fired or laid off for no reason, companies tend to demand loyalty but offer little to nothing in return. So remember that they aren't your friends, don't worry about upsetting them. Just be clear, professional and tell them exactly what is about to happen.
