A lambda sensor is a dual circuit device. You have Low Range, and High Range. Wires are usually 6 in number, 2 low, 2 high and 2 for power supply/earth.
When the vehicle starts and is 'cool' it will use low range circuit - this allows a richer burn, until warm, where the high range circuit takes over and runs much leaner. Low range is effectively telling the ECU to allow the vehicle to have a bit of 'choke' as it were.
If the low range is badly connected, the ECU will be given only info from the high range, thus thinking the vehicle is warm and giving less fuel/air mixture, which could be the problem accompanied by the ECU not being able to 'learn' a new signal.
In most cases of Lambda faults, the signal is intermittent - which causes the 'hunting' and it is usually the high range that fails (as it sees more use) and this gives a richer mixture (as low range is operating), and causing the cat to overheat giving the eggy smell.
Think of it as an electronic version of a thermostat - low is 'closed' and high is 'open'