Lemma 45.9.2. Assume given (D0), (D1), and (D3) satisfying (A) and (B). Let $X$ and $Y$ be nonempty smooth projective schemes over $k$ equidimensional of dimensions $d$ and $e$. Then $\text{pr}_{2, *} : H^*(X \times Y)(d + e) \to H^*(Y)(e)$ sends $a \otimes b$ to $(\int _ X a) b$.

**Proof.**
This follows from axioms (B)(a) and (B)(b).
$\square$

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