Lemma 45.7.6. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ and $Y$ be smooth projective varieties. Then $\text{pr}_{2, *} : H^*(X \times Y) \to H^*(Y)$ sends $a \otimes b$ to $(\int _ X a) b$.

**Proof.**
This is equivalent to the result of Lemma 45.7.5.
$\square$

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