Coverings of small categories and nerves

We prove a certain proposition which states a relationship between coverings of small categories and nerves. As its application, we prove that for a covering $\text{map}PEB$ of finite categories, the zeta function of $E$ is the zeta function of $B$ to the number of sheet of $P$. Moreover, we prove the formula $\chi (E)=\chi (F)\chi (B)$ for Euler characteristic of categories and coverings.