Your mother-in-law has a defective thermal bypass valve. It's this (not fully visible) item in your photo:
The complete valve looks like this:
It may have the name Grundfos or Watts on it. Somewhere in the house there is a hot water recirculating pump, which is usually under a sink or near the water heater. The pump creates a slight pressure difference between the hot water and cold water pipes so that when the above thermal bypass valve is cold (less than 90F), the valve opens and lets hot water into the cold water line.
This brings hot or warm water to the sink so you don't have to wait for hot water, which is the whole point.
When these valves fail, they usually fail in the open position, so they pass hot water all the time, leading to a cold water line full of hot or warm water. When the valve is working correctly, you'll get hot water from the hot side immediately. The cold faucet will give you warm water for a few seconds, then it will be normal cold water.
When the valve is stuck open, you'll get warm water for a long time on the cold water side, then it will turn cold.
These valves are easy to replace. Turn off the hot and cold water, unscrew the two screws from the wall to take the valve off the wall, disconnect the four hoses from the valve and connect them to a replacement valve in the same arrangement as the original. Then screw the replacement valve back onto the wall.
If your mother-in-law doesn't want immediate hot water at this sink and tepid cold water for a few seconds, then cold, the valve can be eliminated. Before she makes that decision, see how long it takes to get cold water now with the defective valve. That's how long it will take to get hot water with no valve. The previous owner probably felt that it was an unreasonably long wait.
To eliminate the bypass valve, connect the hoses that come down from the faucet directly to the hot and cold water valves respectively, eliminating the bypass entirely. (See the valve for the words Hot and Cold if you're not sure which hose is which.) You'll end up with two left-over hoses.
These valves can sometimes be rejuvenated by disconnecting them and soaking in white vinegar for a week or so, but in my experience that only works about half the time.
There may be other sinks in the house with the same type of valve. Also, the circulating pump may be on a timer, so if there's a power failure its clock may need to be reset.
(Photo credit: supplyhouse.com)
