A cat's lethargy and dry nose may be caused by a variety of reasons. Here are some possible explanations:

Environmental factors: Cats may feel comfortable and safe in a quiet environment, which makes it easier for them to fall asleep.

Age: Young or older cats may need more sleep to recover and stay healthy.

Activity level: If your cat has been active recently, they may tire more easily in the future.

Health problems: Certain health problems, such as infection, fever, or respiratory problems, can cause a dry nose in your cat. Additionally, these health issues may also make cats feel tired and lethargic.

Diet and water intake: Make sure your cat has adequate water intake, as a lack of water may cause a dry nose.