I've just completed my 3rd shift as Night Auditor with ease thanks to me actually reading the training manual, but the person who was supposed to be assisting/training/supervising decided, on the two nights that I got to work with her, that she deserved a 3-hour paid nap due to her having two kids. Should I bring this issue up to my manager's attention?
Night shifts vary. A relative who worked as a computer operator (mounting tapes, keeping printers happy) says it was a firing on the spot offense to be found asleep and they were supposed to report each other. A friend who drives an ambulance routinely sleeps through night shifts; they wake up if there is a call. Your manual is unlikely to tell you the truth of the culture at your employer.
Regardless of the workplace culture, if the supervisor's nap is causing you an issue - you're feeling unsafe, you don't know what to do, some things are not getting done because there's too much work for one person - raise those issues with the supervisor without mentioning the sleep part. For example, "sorry to disturb you, but [problem.]" or "Last night, while you were unavailable, [problem.] If that happens again tonight, do you want to be informed right away?"
If you are not having problems, and nobody has told you naps are forbidden, then consider the possibility that the culture of this firm allows night shift people to nap when things are slow. Relax a little and enjoy your job. Getting stuff done and being good at what you do are genuinely pleasant experiences. So is the peace and quiet of a still building overnight, or a glimpse of the moon or the stars through an office window, or being part of a team that all support each other in being good rather than sniping about who is better or more deserving than the other.
To discover the culture, you can ask your coworkers. I'm sure you discussed the nap with the supervisor, since you tell us she explained to you her reasoning. Did she ask you to keep it secret? If so, that suggests it is not approved of. If she did not, that might be because it is fine. You could also try asking a peer - not specifically about the supervisor but in general. "Do people nap on night shifts? Is that allowed?" is a relatively neutral question that could get you useful information. If it's really irking you that the supervisor is napping, and you're sure it's not allowed (it truly is allowed in many on-call situations where the workload varies) then when you go to the supervisor, ask, don't tell. "Are we allowed to nap on the night shift?" for example. And if you get a "no", then "what should I do if I see someone napping?" is a natural followon. And then you will know what your manager wants.
It sounds like you don't need her teaching you because the manual is good. If that is true, do you have any personal disadvantage because of her sleeping?
Then let it be. You gain nothing except for potential problems, hostility etc. Let the manager do his own job (if she does this often, he will notice somewhere in the future).
If her naps disrupt your workflow, then you need to talk to her. Perhaps she can time her naps to minimize disruption to you. Escalating this issue to management is a last resort, especially if no one's safety or good health is involved.
People have different biological clocks. While I can still do 36 hours in 36 hours with ease (I am no longer young), plenty of people can't make it through the third shift without a nap. Better to work with people and their biological clocks than pass judgement on these same people. I am rock solid when it comes to taking physical punishment while working overnight, but I am quite cognizant that not everyone is rock solid and can take punishment.
Look, if her naps don't disrupt your workflow, you otherwise get along fine with her and you expect her to give you a good professional review, what's your complaint?
If she has tasks of her own to complete and she is not completing them because she is napping, that's not your problem.
Yes, you should.
Someone who arbitrarily decides that they are entitled to a three-hour siesta is, in effect, stealing three hours of wages (plus benefits) from the company.
This is much more than a simple lack of effort. This is theft.
Consider what would happen to the company if you too had two children and that you too decided to take a three-hour siesta on the night shift!