Lemma 42.37.7. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. Let $0 \leq p \leq r$. Then the rule that to $f : X' \to X$ assigns $c_ p(f^*\mathcal{E}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \to \mathop{\mathrm{CH}}\nolimits _{k - p}(X')$ is a bivariant class of degree $p$.

**Proof.**
Immediate from Lemmas 42.37.3, 42.37.4, 42.37.5, and 42.37.6 and Definition 42.32.1.
$\square$

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