Let $p: E\to B$ be a covering map, $f: C\to B$ be a continuous onto map between compact manifolds, let $$E'=E\times_B C=\{(c,e)|f(c)=p(e)\}$$ be the pullback with the projection $p': E'\to C$ and the other projection $f':E'\to E$. How to prove that the degree of $f'$ and the degree of $f$ coincide,i.e.,
$${\rm deg}\ f'= {\rm deg}\ f .$$
Could you please give me some help in detail? Thank you very much!