The brief
The A6 had been overheating, the temperature gauge climbing, with coolant puddles under the car and the level dropping between checks. He stopped driving it and brought it in, which is the right call, an engine that overheats shouldn't be run on. Overheating has a few possible causes, a leaking thermostat, a failing water pump, low coolant from a leak, a clogged radiator, so it needs diagnosing rather than guessing. The water pump circulates coolant around the engine and through the radiator to carry the heat away, and when it fails it can leak coolant from its seal, which is the puddle and the dropping level, and it stops moving coolant properly, which is the rising temperature. A hot engine can warp a head, so the cause needs finding and fixing.
The diagnosis
A pressure check and a look at the cooling system traced the overheating to the water pump, which was failing and weeping from its seal. The radiator, the hoses and the expansion tank checked out, so the pump was the fault. That's a pump replacement, with the thermostat in the same circuit done at the same time, fresh seals on both, rather than being back in here when the thermostat lets go.
The work
The cooling system was drained, the failed water pump and the thermostat removed, and new genuine Audi-spec parts fitted with fresh seals. The system was refilled with the correct coolant, the air bled out the proper way, and held under pressure to confirm the seals were dry. A road test confirmed the gauge sat steady and there was nothing dripping.
The outcome
Gauge steady through traffic and at speed, no coolant under the car, no overheating, and the level holding. The A6 went home with the cooling system back to spec. A failing water pump only fails further, and the failure at the end is an overheat that can cost a head gasket, so changing the pump and the thermostat together put the whole circuit right in one visit.