The brief
The A5's driver watched the temperature gauge keep climbing, pulled over, popped the hood, and saw the radiator fans weren't turning. He stopped driving and brought it in, which is exactly right, an engine that's overheating shouldn't be run on. The radiator cooling fans pull air through the radiator when there isn't enough airflow from the car moving, so they do their job in traffic, at idle, and on a hot day. When the fan assembly fails, the motor seizing or the control packing up, there's no forced airflow at low speed, so the coolant can't shed its heat and the engine overheats. A failed cooling fan doesn't recover, so it needs replacing.
The diagnosis
A check of the cooling system traced it to the radiator fan assembly, it wasn't running when it should, the motor failed, so there was no forced airflow at low speed, which is exactly the overheating. The radiator, the coolant, the thermostat and the water pump all checked out, it was the fan. That's a fan assembly replacement, you don't rebuild the fan motor on the car, so the call was a complete cooling fan assembly, fitted and tested.
The work
The bits in the way came off, the failed radiator fan assembly removed, and a new genuine Audi-spec cooling fan assembly fitted, connected up, every fastener torqued to spec. With the engine warmed up the fans were checked to cut in at the right temperature and pull proper airflow through the radiator. A road test, including a spell of low-speed running, confirmed the gauge sat steady with no overheating.
The outcome
Gauge steady in traffic and at idle, the fans cutting in when they should, and no overheating at low speed. The A5 went home with the cooling sorted. A failed cooling fan only leaves the engine to overheat every time you're stuck in traffic, and an overheat can cost a head gasket, so changing the fan assembly put it right before it did damage.