**Core Concept**
The cough response is a protective reflex that occurs when the cough center in the medulla oblongata is stimulated. This reflex is mediated by the trigeminal nerve, which has branches innervating the external auditory meatus, and the vagus nerve, which has branches innervating the larynx and lungs.

**Why the Correct Answer is Right**
When the ear canal is cleaned, the stimulation of the auricular branch of the vagus nerve (specifically, the Arnold's nerve) can activate the cough reflex. This is because the vagus nerve contains sensory fibers that are sensitive to irritation, and when these fibers are stimulated, they can send signals to the cough center in the medulla, triggering a cough response. The cough reflex is an important protective mechanism that helps to clear the airways of foreign particles and irritants.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because the trigeminal nerve is involved in facial sensation and motor functions, but it is not directly responsible for mediating the cough response in the ear canal.
**Option B:** This option is incorrect because the glossopharyngeal nerve is involved in swallowing and taste, but it is not directly responsible for mediating the cough response in the ear canal.
**Option C:** This option is incorrect because the facial nerve is involved in facial motor functions and taste, but it is not directly responsible for mediating the cough response in the ear canal.

**Clinical Pearl / High-Yield Fact**
The Arnold's nerve, a branch of the vagus nerve, is responsible for transmitting sensory information from the external auditory meatus to the brain. This nerve is often overlooked in medical education, but it plays a crucial role in the cough reflex and other protective mechanisms.

**Correct Answer:** C. The vagus nerve.