Lately I've noticed, after the completion of coitus, I become very sleepy. Can I be developing narcolepsy?

Probably not. Narcolepsy would manifest itself differently. It is common for somnolence to occur after sex as a consequence of relaxation and exertion.
Normal reaction. Most "normal" people feel a little bit tired after sex/coitus, especially if the act is relatively involved (eg lasts longer than 5 minutes). Think of it as a heavy physical workout, and you'll understand why that might be the case. Narcolepsy, on the other hand, causes sleepiness at random times during the day with or without exercise. Good luck and stay healthy!