Notes on the little wrench and nozzles in general...
I do a lot of restoration work and repair. The single largest airbrush killer out there is beginners trying to tighten a nozzle, with the little wrench, or even god forbid, pliers.
My advice for anyone new to airbrushing.... Under no circumstances, save the most dire, ever, ever, ever, ever remove, tighten or alter in any way whatsoever the nozzle. Even most clogs can be removed with a good soak and an interdental brush or sharp toothpick GENTLY applied from the back end, and even then, never ever, ever do that if you can see your needle tip coming through the nozzle. Even stubborn paint chunks of that tiny, tiny size will soften and blow out after an overnight soak in cleaning fluid or a mix of warm water and dishsoap.
It is common when starting out to think of the nozzle as needing independent cleaning, or being a source of air leakage, but this is simply not true. Nozzles pretty much stay reasonably clean in regular use and with proper back flushing or spraying cleaner through the brush. The ONLY time you EVER remove a nozzle is when something very, very solid gets stuck in there, or when you need to replace it because of physical damage, and in general, that should probably only be done by someone with experience who knows that you never tighten them on with anything more than single finger pressure. Once a nozzle is installed on the fluid head, that is it. Do not remove.
When installing a new nozzle, use the little wrench or a custom tool for it. Never, ever, ever use pliers or anything else. Nozzles are a softer metal that won't fuse with the brass body over time like aluminum or hardened steel can (chemical reaction that can happen over years). The slight give to them means that needles can make a good seal, preventing paint leakage. They are absolutely not meant to take any stress whatsoever. So when you use the little wrench to put on a brand new nozzle, use your little finger only, to turn the wrench and immediately stop the very instant you feel a bit of resistance. Never, ever, ever, ever, ever, "tighten".
If you have reason to believe that the nozzle is leaking or air is being forced back up the paint channel through a leak (bubbles in the cup usually), use a bit of chapstick or beeswax to seal them. Do not tighten.
I know I keep saying that "do not tighten" over and over again like a broken record player, but it is because it is responsible for more than 95% of all "broken" airbrushes I get in the shop. It is incredibly common to have them come in when some well intentioned new user has unnecessarily disassembled the nozzle for cleaning, then reinstalled it, and tried to make it air tight, and then ended up breaking the nozzle off, leaving a ring of threads in the body. A real mess to sort out.
Airbrush nozzles are delicate precision machined parts. Not robust mechanical parts. They need to be treated like delicate jewelry. It is best to never, ever take them off, as it will vastly extend the life of your airbrush.
Also, note, that they have nothing to do with air coming out of the gun. They control the amount of paint that mixes into the airflow. Usually if paint isn't flowing, it is because the paint mix is too thick for the size of the nozzle or pressure. So when testing your spray, start with water. See if it comes out in a nice cone. Trouble or damage with the nozzle will usually manifest as bubbles in the cup (air is being allowed back up into the paint channel due to a crack or some other issue with where the nozzle meets the needle). Chugging or erratic spray is nearly always some kind of air leak, and remember.... the nozzle really has nothing to do with the air flow.
So, unless the nozzle has been mashed, dropped on it's tip or something else, you should avoid taking that nozzle off like it was the plague. Once it is on, it stays on. Forever. Or at least until it gets damaged and it's time for a new one.
I learned all of this the hard (and expensive) way over the years. I pass it on to save new airbrush folk a lot of pain, frustration and an awful lot of money.
Never remove or tighten. If you have to install a new one, little finger only, and DO NOT TIGHTEN, just turn until the first bit of resistance.
Good luck!