The correct answer is D) at any time when underway. According to the International Regulations for Preventing Collisions at Sea (COLREGS), a pilot vessel on pilotage duty is required to display sidelights and a stern light at any time when it is underway, regardless of whether it is operating at night or day. This is because these navigation lights are essential for ensuring the visibility and identification of the vessel to other vessels in the vicinity, which is crucial for maintaining safe navigation. The other options are incorrect because: A) the sidelights and stern light must be displayed at all times when underway, not just when the identifying lights are not being shown; B) the requirement to display the navigation lights is not dependent on whether the vessel is making way or not; and C) the requirement to display the navigation lights applies to vessels that are underway, not at anchor.