**Core Concept**
The cerebellar cortex is responsible for coordinating motor activities, including posture, balance, and fine motor movements. It receives input from the proprioceptive and vestibular systems and sends output to the brainstem and spinal cord to regulate motor responses.

**Why the Correct Answer is Right**
The cerebellar cortex sends output through the cerebellar nuclei, which then project to the brainstem and spinal cord. The cerebellar nuclei are the only output structures of the cerebellar cortex, and they play a crucial role in regulating motor movements. The cerebellar nuclei receive input from the Purkinje cells, which are the primary output neurons of the cerebellar cortex.

**Why Each Wrong Option is Incorrect**
**Option A:** The inferior olivary nucleus is an input structure to the cerebellar cortex, not an output structure.
**Option B:** The dentate nucleus is one of the cerebellar nuclei, but it is not the only output structure of the cerebellar cortex.
**Option C:** The fastigial nucleus is another cerebellar nucleus, but it is not the sole output of the cerebellar cortex.

**Clinical Pearl / High-Yield Fact**
The cerebellar nuclei are involved in the regulation of motor movements, and damage to these structures can result in ataxia and other motor coordination disorders.

**Correct Answer:** C. The cerebellar nuclei.