Mix and air combine in a mixing chamber of a housing of a frozen product system to form a mixture. A cap received in one end of the housing includes an air inlet, a mix inlet, and two mixture outlets. An integral orifice received in the air inlet regulates the amount of air that flows into the mixing chamber. A gasket positioned adjacent the cap provides a liquid tight seal that prevents air and mix from entering the mixing chamber when the mixture is dispensed from the mixing chamber and to prevent the mixture from exiting the freezing cylinder while air and mix are drawn into the mixing chamber. The gasket includes an air valve, a mix valve, and two mixture valves. An adapter draws air into the pump through an air hole and draws mix into the pump through a mix hole. A piston slides in the mixing chamber to dispense the mixture through mixture holes. The cap includes a plurality of protrusions that engage holes in the adapter to ensure proper alignment of the cap and the adapter. The gasket also includes a protrusion that engages a cavity in the cap to ensure proper alignment of the gasket with the cap.