Hotelling’s Continuous Model - game theory
Pick an arbitrary cdf $F$ that is supported on $[-a,a]$. The median, $m$, must satisfy $$\int_{-a}^{m}dF \geq \frac{1}{2}, \quad \text{and} \quad \int_{m}^{a}dF \geq \frac{1}{2}$$

Note that $m$ is not generally $0$.

The phrase you mention, "the unique NE is where both candidates chooses the policy associated with the median voter," simply means that the unique NE is where both candidates choose a policy located on $m$. 

This is easy to show, think of "unraveling toward the center."