Let $M$ and $N$ be smooth manifolds, let $f:N\to M$ be a smooth map, and let $\pi:E\to M$ be a smooth complex vector bundle of finite rank $r$. Let $f^*E\to N$ denote the pullback complex vector bundle, and let $f^*:H^{2k}(M;\mathbb Z)\to H^{2k}(N;\mathbb Z)$ denote the pullback homomorphism in singular cohomology. For every integer $k\ge 0$, the $k$-th Chern class satisfies

\begin{align*} c_k(f^*E)=f^*c_k(E) \end{align*}

in $H^{2k}(N;\mathbb Z)$, with the convention that $c_0(E)=1\in H^0(M;\mathbb Z)$ and $c_k(E)=0$ for $k>r$.