Captain reported cabin failed to pressurize during climb. During troubleshooting it was noticed that the cabin pressure source knob was in the off position. Flight crew moved the switch to both position; restored cabin pressure and continued to destination airport.

Narrative

A quick summary is that turning on the cabin pressure source knob got missed during the after-start checklist; which resulted in a Red CABIN PRESS LO Warning during climb out; followed by the release of the cabin oxygen masks. Out of ZZZ; we were climbing through 16;000 ft. for FL 200 when the red CABIN PRESS LO Annunciator came on. I leveled off and we asked ATC to descend back down to 10;000 ft. and to return to ZZZ; and I told the Pilot Monitoring (PM) to get out the QRH and run the checklist. ATC asked if we were [requesting priority handling] and we said no; as at that point our cabin pressure was showing just above 10;000 ft. and was not climbing; and we were not in the flight levels. The PM started on the QRH checklist and this was when I noticed that the cabin pressure source knob was still turned to OFF. We turned it to BOTH and the cabin pressurized normally. The red annunciator went off; and the cabin altitude and differential returned to normal. The PM and I discussed the situation; considered whether an air return was necessary/advisable; reviewed our own system and standards knowledge; and decided that the pressurization situation was now normal and there was no cause for further deviation. We talked with our passengers (who expressed that they were not alarmed or concerned); and then the crew decided to continue on to the final destination of ZZZ1. The flight then went normally. Oxygen gauge remained green throughout. There are two places that this issue should have gotten caught. One is during the after-start checklist; where it indicates that the SIC should set the cabin pressure source knob to BOTH NORM and then set the altitude in the cabin controller. The second is during the climb check below the line; where it says; 'Cabin Pressurization...CHECK.' This checklist is run silently by the PM at 10;000 ft.; and at that time the cabin pressure (long needle) should be close to sea level and the differential should be around 4 PSI. Any other reading should be noticed and a source of crew discussion.A potential issue is that when the SIC is also the PM; at no point does the left seat pilot acknowledge the pressure settings. This could be corrected on the checklist; by adding a call and response during the BEFORE TAXI checklist. This would not be unprecedented as there are several confirmation steps already on the BEFORE TAXI checklist (FMS set for departure; Avionics/altimeters; V speeds; etc.). I'd suggest putting this as a new #13; after 'Rudder Boost;' add 'Cabin Pressure System...CONFIRM BOTH NORM AND SET.'