The end of a cat's nose should be dry under normal conditions and good health in my considered viewpoint subject to condensation (see below). Over the years, there has been quite a lot of discussion about the end of a domestic cat's nose and whether it should be dry or wet or damp or whatever else it should be. I think people are getting confused about it.
It depends upon the external conditions. If you take an indoor cat who is healthy and on the assumption that the ambient temperature in the home is comfortable for human and cat then the end of that cat's nose should be dry. If that cat then goes outside into cold winter weather the end of her nose may become damp because of condensation from her damp breath forming on the end of her nose.
Temporarily her nose will be damp but this is due to physics. It has nothing to do with her health. If she is unhealthy with a cold then she may have a runny nose so the end of her nose will be damp. If she is a little confused or stressed she may engage in what is called "displacement activity". This may entail her licking the end of her nose repeatedly. This would make her nose damp.
Dr Yuki Hattori, a well-known Japanese veterinarian, in his book What Cats Want states incorrectly in my view that the end of a cat's nose should be damp. I genuinely believe that he is confused about this. He argues that it is damp so the cat can smell scent molecules more effectively. That I just cannot believe. Yes, certainly if scent molecules are damp they can be smelled more effectively but this is not a reason why the end of a cat's nose should be damp.
My cat's nose is dry. I've just touched it. He's inside with me in dry, warm conditions. His nose reflects those ambient conditions. I would challenge anybody who reads this (and they will be very few people who do!) to carry out their own test and touch the end of their cat's nose. You can then tell me in a comment whether it is dry or wet or something in between such is damp.
To reiterate, the dampness or dryness of the end of your cat's nose depends upon a range of circumstances and conditions one of which is the ambient air temperature. You cannot discuss the end of your cat's nose in isolation of ambient air temperature and a cat's health. But the default situation should be a dry nose leather in my view. The phrase "nose leather" is a cat fancy term meaning the end of the cat's nose which looks a bit like leather. As the flat bit out ofwhich two nostrils protrude.