Technically you are both right. Thats actually an axial flow turboprop jet engine.
The propellor attaches to the shaft to the left there, and serves the same function as the primary intake fan on your typical bypass turbofan jet engine. The air is then compressed further by a stack of turbine blades called a compressor, fuel is injected, the entire air fuel mass is ignited, then ran through a turbine designed to scavenge power from the exhaust, which in turn turns the compressor. THe thrust exhausts out the back and ads to the total trhust of the engine.