Lemma 45.5.2. Let $k$ be a base field. The functor $\mathop{\mathrm{CH}}\nolimits ^ i(-)$ on the category of motives $M_ k$ is representable by $\mathbf{1}(-i)$, i.e., we have

\[ \mathop{\mathrm{CH}}\nolimits ^ i(M) = \mathop{\mathrm{Hom}}\nolimits _{M_ k}(\mathbf{1}(-i), M) \]

functorially in $M$ in $M_ k$.

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