A moist nose: A healthy cat's nose should ideally be cool and moist, as this helps maintain its keen sense of smell. Smell is extremely important for cats, and moist air makes it easier for them to detect scents.
Dry nose in cats can be caused by various factors, including constipation, fever, colds, and illness-related heat. If you notice your cat's nose is dry, check for other abnormalities and seek veterinary care as soon as possible. 
Environmental factors: A dry nose in cats does not always indicate illness. If a cat is in a hot, dry environment, even if it drinks plenty of water, its nose will have difficulty staying moist. In this case, a dry nose is unrelated to disease.
When resting or sleeping: A cat's nose will be dry when it is resting or sleeping. This is because a cat's body temperature remains constant and its body is in a stable state when sleeping, which leads to a dry nose. When the cat wakes up and moves around a bit, its nose will return to a moist state.
What to do if your cat has a dry nose: If you notice your cat's nose is dry, first check if it has symptoms of illness such as fever, loss of appetite, vomiting, or diarrhea. If not, you can rule out illness. Next, check if your cat is in a hot, dry environment (such as on a balcony in summer). If so, do not allow your cat to be in a hot environment for extended periods, as this could lead to heatstroke.
Seek veterinary treatment promptly: If the condition is confirmed to be caused by illness, take the cat to the vet for treatment as soon as possible. After treatment, once the cat recovers, the dryness of its nose will naturally improve and become moist again.