Lemma 45.4.1. The category $M_ k$ whose objects are motives over $k$ and morphisms are morphisms of motives over $k$ is a $\mathbf{Q}$-linear category. There is a contravariant functor

defined by $h(X) = (X, 1, 0)$ and $h(f) = [\Gamma _ f]$.

Lemma 45.4.1. The category $M_ k$ whose objects are motives over $k$ and morphisms are morphisms of motives over $k$ is a $\mathbf{Q}$-linear category. There is a contravariant functor

\[ h : \{ \text{smooth projective schemes over }k\} \longrightarrow M_ k \]

defined by $h(X) = (X, 1, 0)$ and $h(f) = [\Gamma _ f]$.

**Proof.**
Follows immediately from Lemma 45.3.4.
$\square$

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