I could NOT find anything with a quick search on P013C, so you want to confirm what it means.
It is possible they replaced the wrong O2 sensor.
Its also possible the problem is NOT the O2 sensor, but something related to the O2 sensor, thats why it keeps setting the code, even though they've replaced the O2 sensor multiple times.
O2 sensor slow to respond, that could mean:
- The O2 sensor is bad
- There could be something wrong with the wiring of the O2 sensor
- There could be something wrong with the engine creating exhaust that isn't making the O2 sensor respond as quickly as it should.
- There could be something wrong with the Catalytic Converter, which the downstream O2 sensor is there to monitor the output of the CAT, that cause the O2 sensor to respond to slowly.
The Dealership should have Factory Service Manuals with specific troubleshooting procedures to narrow down the actual cause to fix it, instead of just performing the most obvious replacement again and again, with the same result, which is failure.
An independent shop may NOT be able to support individual service manuals for your specific vehicle, but something like this is fairly generic in operation to other vehicles, they should have some sort of trouble shooting guide.
Yes, the Dealership should have the advantage over independent shops, with factory service manuals and special tools. BUT, sadly, the Dealerships have been so insulated from competition for various reasons, they have no incentive to maintain any quality and despite having all the advantages, they still can't do as good of a job as an independent shop. The manufacturers have even, for all practical purposes, locked DIY'er and Independent shops out of a lot of the elctronics in your vehicle forcing you to the dealership as the only one to be able to perform the procedure to repair your vehicle.