In this case, the pump pushes hot water to a Watts (brand) thermo valve under the sink. Until the hot water reaches the valve it allows the "cold" hot water to recirc into the cold line. When the valve reaches its set (?) temp it closes the cold side off, and the system acts normally. Yes, some warm water enters the cold side as the valve warms, but that really doesn't matter unless you don't like brushing your teeth with lukewarm water!!
Without the valve, with just a pump and a cross link between the lines, you will find either side will mix to the open sink valve. Path of least resistance, they will mix, Mex!!
I and others do this to save water out in the boonies, instead of sending a few cups or quarts of unused water into the gray tanks waiting for the shower or sink water to warm up. My bath is aboot 20 ft + from the water heater, and a bunch of water just goes down the drain that can extend our stay and tank fill up times. Not really needed when hooked up, just can let the water go down the drain.
Last trip, 26 degs in am, loved having hot water, and saving water. Enclosed and "heated" bottom of trailer only keeps lines from freezing to a point, doesn't warm the shower water!!!