  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 $\map{P}{E}{B}$ 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.
