Pilot reported cabin pressure and door warning lights illuminated during departure climb. Flight crew returned to departure airport and maintenance found a door was not properly closed.

Narrative

This was a Part 91 ferry flight from to reposition the aircraft to ZZZ1 for an APU repair. (Only two pilots onboard.) There was a problem finding equipment to start the engine; finally the Mechanic found an air start and a GPU. We started the engines and the Mechanic told us he was leaving the aircraft using the aft exit door and closing the access ramp from the outside. During climb out at approximately 10;000 ft.; the 'flow' light and 'cabin press' lights came on followed by the 'aft cabin door.' We returned to ZZZ and the Mechanic found the door not properly closed. We did not have any lights prior to takeoff.Mechanic closed the door; and we the pilots; failed to go back and check if the door was properly closed; since we did not have any warnings; we thought or assumed the door was locked. There was a lot of pressure from the airline to move the aircraft to ZZZ1 ASAP because it had a flight scheduled once the APU repair was completed. No flight attendants onboard. They normally open and close all doors. We verify lights are out!