If your tech thinks a dead or weak battery can contribute to cat or O2 sensor codes he doesn't have much understanding of the OBDII system. If someone else is available they might be a better bet. At least start over with whoever you choose and get a correct reading of which codes are actually set, then look them up.

P0420 is a catalyst efficiency code and can be caused by a bad cat if nothing else is wrong. It may show up (with other codes too) if the engine is running incorrectly, e.g. too rich.

If you receive codes related to either O2 sensor they are usually more specific and will point to which sensor.

However as Klaus will tell you various vacuum or exhaust leaks can cause O2 or cat related sensor codes so those should be eliminated systematically first.

good luck!