Final answer:
Droopy eyelids and double vision are typically a result of an injury to the oculomotor nerve, which is responsible for controlling most of the eye's movements and the levator palpebrae superioris muscle that elevates the upper eyelid.
Step-by-step explanation:
Droopy eyelids and double vision can result from injury to the oculomotor nerve. The oculomotor nerve, which is the third cranial nerve, controls most of the eye's movements, including the levator palpebrae superioris muscle responsible for elevating the eyelid. When this nerve is damaged, it can lead to a condition known as ptosis, which is characterized by drooping of the eyelids, as well as diplopia or double vision, due to the misalignment of the eyes.
On the other hand, the abducens nerve, which is the sixth cranial nerve, controls the lateral rectus muscle that is responsible for abducting the eye. The trochlear nerve, the fourth cranial nerve, is responsible for the superior oblique muscle, which rotates the eye. While these nerves also play roles in eye movement, they do not control the eyelids.