What happens if I only install the check valve in the futurist hot and cold sink? And not install the pump?

Nothing would happen the sensor valve only gives the system a point to recirculate water to and from the pump is needed to increase the water pressure on the hot side in order to push the stagnant water into the cold water side and bring hot water up to the sensor valve. In short you have to use both components you cannot use one without the other.