Lemma 45.4.9. Let $X$ be a smooth projective scheme over $k$ which is equidimensional of dimension $d$. Then $h(X)(d)$ is a left dual to $h(X)$ in $M_ k$.

**Proof.**
We will use Lemma 45.3.1 without further mention. We compute

Here we have $\eta = [\Delta ]$. On the other hand, we have

and here we have the class $\epsilon = [\Delta ]$ of the diagonal as well. The composition of the correspondence $[\Delta ] \otimes 1$ with $1 \otimes [\Delta ]$ either way is the correspondence $[\Delta ] = 1$ in $\text{Corr}^0(X, X)$ which proves the required diagrams of Categories, Definition 4.43.5 commute. Namely, observe that

is given by the class of the cycle $\text{pr}^{1234, -1}_{23}(\Delta ) \cap \text{pr}^{1234, -1}_{14}(\Delta )$ with obvious notation. Similarly, the class

is given by the class of the cycle $\text{pr}^{1234, -1}_{23}(\Delta ) \cap \text{pr}^{1234, -1}_{14}(\Delta )$. The composition $(1 \otimes [\Delta ]) \circ ([\Delta ] \otimes 1)$ is by definition the pushforward $\text{pr}^{12345}_{15, *}$ of the intersection product

which is equal to $\Delta $ as desired. We omit the proof of the formula for the composition in the other order. $\square$

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