**Core Concept**
The medulla oblongata is a critical part of the brainstem that plays a vital role in controlling various autonomic and motor functions. The lateral part of the medulla is involved in the transmission of sensory and motor signals to and from the spinal cord and other parts of the brain. A lesion in this area can lead to a range of neurological deficits.

**Why the Correct Answer is Right**
The correct answer is **B. Contralateral loss of proprioception to the body and limbs** because the lateral part of the medulla contains the crossing fibers of the medial lemniscus, which carries proprioceptive information from the spinal cord to the brain. A lesion in this area would disrupt these fibers, resulting in contralateral loss of proprioception. This is because the fibers of the medial lemniscus cross to the opposite side of the brainstem at the level of the medulla, so a lesion on one side would affect the other side of the body.

**Why Each Wrong Option is Incorrect**
**Option A:** Ipsilateral Homer's syndrome is incorrect because a lesion in the lateral part of the medulla would typically cause contralateral Homer's syndrome, not ipsilateral. This is because the sympathetic fibers that control the pupil and eyelid are located in the lateral medulla and cross to the opposite side at the level of the medulla.

**Option C:** Nystagmus is incorrect because while a lesion in the lateral part of the medulla can cause nystagmus, it is not a characteristic feature of lesions in this specific area. Nystagmus can be caused by lesions in other parts of the brainstem, including the midbrain and pons.

**Option D:** Dysphagia is incorrect because while dysphagia can be caused by lesions in the medulla, it is not a characteristic feature of lesions in the lateral part of the medulla. Dysphagia is more commonly associated with lesions in the nucleus ambiguus or the hypoglossal nerve.

**Clinical Pearl / High-Yield Fact**
A key point to remember is that the lateral medullary syndrome (also known as Wallenberg's syndrome) typically presents with a combination of ipsilateral Homer's syndrome, contralateral ataxia, and dysphagia, but not contralateral loss of proprioception. This is because the lesion typically affects the spinal trigeminal nucleus and tract, which are responsible for the ipsilateral Homer's syndrome and other symptoms.

**✓ Correct Answer: B. Contralateral loss of proprioception to the body and limbs**