Let $R$ be a commutative ring with unit. Let $r\ge 1$ be an integer, and for each $i\in\{1,\dots,r\}$ let $d_i\ge 0$ be an integer. Suppose that $c_i$ is a characteristic class assignment for smooth real vector bundles with coefficients in $R$, so that for every smooth manifold $X$ and every smooth real vector bundle $E\to X$ one has

\begin{align*} c_i(E)\in H^{d_i}(X;R), \end{align*}

and suppose that $c_i$ is natural in the sense that, for every smooth map $g:Y\to X$ and every smooth real vector bundle $E\to X$,

\begin{align*} c_i(g^*E)=g^*c_i(E). \end{align*}

Let $M$ and $N$ be closed oriented smooth $n$-manifolds, and let $f:M\to N$ be an orientation-preserving diffeomorphism. Let $P\in R[X_1,\dots,X_r]$ be weighted homogeneous of cohomological degree $n$, where the variable $X_i$ has degree $d_i$. Then

\begin{align*} \left\langle P(c_1(TN),\dots,c_r(TN)),[N]_R\right\rangle=\left\langle P(c_1(TM),\dots,c_r(TM)),[M]_R\right\rangle \end{align*}

in $R$, where $[M]_R\in H_n(M;R)$ and $[N]_R\in H_n(N;R)$ are the fundamental classes determined by the orientations.