My kitchen sink has a peculiar setup I haven't seem before. The faucet has 3 pipes and there is an electric water heater under the sink. The first pipe brings cold water into the faucet. The second pipe bring cold water from the faucet into the water heater. The third pipe brings warm water from the heater back to the faucet. Why can't the cold water be supplied directly to the water heater without the initial trip to the faucet?
This is known as a vented tap.
The valve part of the hot water tap operates on the mains pressure cold water inlet to the heater.
The outlet part of the hot water tap is vented i.e. always open, so the hot water can expand and relieve pressure without restriction.
Because these local water heaters are often open systems, i.e. the increased pressure from heated water must not damage the installation. The pressure of the water net is only up to the faucet valve. After that valve it is in contact with the normal air (pressure) and flows to the heater, back to the faucet valve where it can be mixed with the cold water. Many countries have rules that heated water must not flow back to the cold water net, a backflow preventer is in the incoming line of the heater. That is why the pressure can not be decreased upstream, only downstream. It is dangerous to use a normal faucet with those heaters, since the hot water tank could burst spreading hot water into the room.