Let $X$ be a nice space (compact $CW$-complex or triangulated space, compact manifold, whatever works), $f:Y\to X$ be a finite covering of degree $n$, and $\chi(X)$ be the euler characteristic. By the Lefschetz fixed point theorem (or by counting cells of $X$ and $Y$ with the induced cell structure) one obtains

$$\chi(Y)=n \cdot\chi(X)$$

I wonder if there is a sheaf theoretic proof of this statement, or even better a sheaf theoretic generalization of this statement by replacing the constant sheaf by any other local system.