If one makes a paste and lets it dry, it is better than adding water that just causes it to soak through. The white vinegar with the soda is less likely to soak in so far. I used it even on expensive Black and white wool carpet with the best results.
Baking Soda is a good product to get rid of oders, that is why a box in the refridgerator works.
But why mix them? Is sodium acetate good at absorbing odors? Because that’s what you get.
I dunno, had a Jesse Pinkman moment and firgure the two would go well together. Hence: I’m not a real chemist.
This. You can also see on the products page that they sell syringes for carpet injection so the product can get in and soak the padding, too. It’s an acceptable fix until the carpet can be replaced, and it may take at least a gallon if not two to do the job.
Once it’s time to replace the carpet, he can assess whether the urine has reached the wood underneath. If it’s pressboard sub-floor type stuff it may be inexpensive enough to just replace it. If it’s wood flooring, sealants can be used prior to re-carpeting.
Something I need to address, in the OP it’s mentioned there are four cats but “the litterbox.” If there really is only one litterbox for four cats, I’m sorry but your friend is kind of an idiot. The rule for litterboxes and cats is one for each cat plus one. He should have five litterboxes. I often forgive people that one extra if they are very large ones - I have four cats and four boxes, but they are in two different rooms and are very large and high-sided, have a few inches of litter in them, and are scooped every day. People with too few litterboxes will always, always, always have issues, and it’s not the fault of the cats.
People who tell you any product (yes, including Nature’s Miracle) actually works either:
a) are lying
b) have a terrible sense of smell
or
c) weren’t dealing with very much cat urine
We had cats who peed in our carpeting repeatedly. A brand new house. I poured gallons of Nature’s Miracle on the stuff. Then I tried lots of other products. Nothing helped. When it was time to sell the house we had to rip out the carpeting, the pad, the floor molding, sand the floorboards and paint them withKillz before replacing all of the above. Guess what? On a humid day, I can still smell the cat pee. (I have a very sensitive nose.)
I will never, ever have a cat again.
The idea of a paste is to allow the products to work and not soak the carpet. I used it many times and it worked so I didn’t dispute it. Vinegar gets rid of the stain and Soda the oder.
Middle school physical science anyone? Acids and bases?
When you mix them, you no longer have either reagent. Have you never made a volcano this way as a child? By “paste” do you mean “frothy mess”?
HOAc (vinegar) + NaCO3H (baking soda) –> CO2 + H2O + NaOAc (neither vinegar nor baking soda)
Oh, good, it’s not just me - I was wondering the same thing.
I used more Soda than vinegar, there is no mess, I just spread it out let it set and when dry vacuumed it up, Why it worked I don’t know,just I had read this many years ago; and when I did cleaning I cleaned some very expensive carpeting and it worked, that is all I can tell you.It took the stain out of some very expensive black and whit wool carpet left no stain or oder.
The recommendation above to rip out the carpet, dry the stairs, and then apply Kilz or similar sealant is dead on, but it omits the critical first step: Get rid of the cats. 
The cats were both dead and gone by that point.
But their smell lingers on still.
One correction.
The owner can NOT assess whether it has reached the wood… The owner is clearly desensitized.
first thanks to all who posted here - I have been trying to research what to do and there is a lot of good info.
My situation - we have an “elder dog” and a two story house. She has trouble with her back legs which have caused her to have problems knowing when she needs to go. The scary thing is the “solid stuff” is much less of a problem than the pee.
Years ago in another house we had a cat with kidney problems and I know that cat pee is much worse than dog so in that respect, even though our dog is 75 lbs, there just isnt that “unique” odor with cat pee.
my 2 issues were the “toxicity” of the air ie when we did try to clean I would feel very nauseous, and second, how to remove the smell.
She has “hit” the stair landing a few times and even with hand cleaning using a wide variety of cleaners as well as having our carpets professionally cleaned on a periodic basis, the “lingering” comes back.
As we do have an elder dog and there is probably a good chance that when she is gone we will get another dog/puppy replacing or pulling the carpet up just isnt an issue/option for us right now.
So a couple questions for the “experts”:
- we have neutral beige carpet - will the vinegar stain this (not that I really care that much at this point) and do you use STRAIGHT vinegar.
- do you need to hand dry - I know my understanding is that vinegar will also remove mold issues but again dont want to heap one thing on top of another and wonder how long will take to dry.
- do you think if I buy some rubber backed area rugs ie large bathroom mats, that the dog will still be able to “smell” and try to hit the area again?
Its funny but we have pergo type flooring downstairs and if I could would have put this stuff ALL over the house as it is super low maintenance. (not sure if you can do stairs with it?).
Again thanks to this board - I did so much research and this was the place that had the most varied responses.
ANOTHER note on the OP’s comment, I have a friend who has 7 indoor/outdoor “rescue” cats and she has NO cat smell - she has I think 10 litter boxes which she keeps immaculately clean so it definately CAN be done. Granted she has a large house and doesnt have a full time job but it can be done if you are diligent.