No hydraulics here, all pneumatic. That cylinder does look a lot like a hydraulic cylinder though. The system operates by two air sources. The first source is direct from the tank and is plumbed into the filter / regulator / lubricator then through a control valve and ultimately to the cylinder. The control valve is an air operated spring return affair that simply switches one side of the cylinder to the air source and the other to exhaust, and flips the sides when activated. The weight of the arm is such that it will provide enough tension on the belts to allow starting the compressor with no air in the system. As air builds in the tank the cylinder is defaulted to pull down on the arm. It only takes about 5 to 10 psi in the cylinder to move the arm to is lowest position. The idler is spring loaded to provide even tension and a small range of over travel for the arm. This side of the system is set at 30 psi, partly to control noise and to ensure enough force is available. Once the tank pressure rises above 30 psi the regulator keeps the cylinder pressure at 30 regardless of the tank pressure. Cylinder speed is controlled by a combination of exhaust flow control - mufflers, and this particular cylinder also has an end cushion on the downward stroke that further slows the last inch or so of the downstroke. Without these the arm movement is pretty much instant - pretty scary in my opinion.
The 2nd air source is the control. It is piped from the pilot unloader valve through a small regulator and into the control port on the air/spring control valve. If this compressor had an engine mounted on it, this is the same signal line that usually works the engine throttle to idle the engine when the pump unloads. The regulator in this line is dual purpose, as it can be used to disable the arm function to provide continuous pump run operation by turning the output pressure down to 0, and knocks the tank pressure signal down from 170 to something in the valves control port rated operating range of 45-150 psi when in clutch mode. I usually use 60 psi.
Clutch operation is as follows: when the tank pressure reaches 170 psi, the pilot unloader valve trips, supplying tank pressure air to the compressor unloaders and also to the control pipe. The control valve is flipped by the presence of air pressure on its port, and reverses the supply and exhaust to the opposite sides of the cylinder causing it to extend, and the belts get loose and the pump stops. Once the tank pressure drops down below 140 psi the pilot unloader valve resets, the control air pressure drops to 0, and the compressor unloaders are set to load (except on this compressor they won't load without lube oil pressure present in the pump as well), and the spring returns the control valve to its normal state, retracting the cylinder and tightening the belts and starting the pump.
The clutch can be manually actuated by the flip tab on the pilot unloader valve for starting with air already in the tank or to prevent the compressor from starting automatically while leaving the engine running. When starting with little or no air in the tank, the clutch remains engaged, even with the flip tab set on the pilot unloader. This is because the control valve will not overcome the spring pressure in the control valve until the tank reaches about 20 psi. This hasn't been an issue for starting as the compressor isn't pumping against any tank pressure, and the unloaders will start to work when air begins building.