Let $f:N\to M$ be a smooth map of smooth manifolds, let $E\to M$ be a smooth complex vector bundle of rank $r$, and let $\nabla$ be a smooth connection on $E$. Let $f^*E\to N$ be equipped with the pulled-back connection $f^*\nabla$. If $P$ is an $\operatorname{Ad}$-invariant complex polynomial on $\mathfrak{gl}(r,\mathbb C)$, evaluated on curvature forms by the standard Chern-Weil convention, then

\begin{align*} P(F_{f^*\nabla})=f^*P(F_\nabla) \end{align*}

as complex-valued differential forms on $N$. Consequently,

\begin{align*} [P(F_{f^*\nabla})]=f^*[P(F_\nabla)] \end{align*}

in $H^{\mathrm{even}}_{\mathrm{dR}}(N;\mathbb C)$.