Consider a sealed box, well-insulated on all sides, except for the lid which is transparent to infrared. An object is placed inside the box and the box is evacuated (purpose being to thermally isolate the contents of the box from its surroundings). The box is placed outdoors (in an everyday atmosphere) on a clear night. Let's assume that at the start of this experiment, the box and its contents are in thermal equilibrium with its surroundings. The object inside the box will radiate infrared according to its temperature, which should escape through the lid of the box. With nothing but clear dark sky above, I assume there is nothing to radiate appreciable heat back into the box and maintain the object's temperature.
Question: will the object cool below the ambient temperature outside the box?