I can't prove this lemma in "Notes on motivic cohomology, Beilinson, Macpherson, Schechtman":
Lemma. A pure functor is exact.
Definitions: A mixed category $\mathcal{M}$ is a $\mathbb{Q}$-abelian category in which every object has a finite increasing filtration $W_\bullet$, such that morphisms are strictly compatible with $W_\bullet$, graded quotients $\mathrm{gr}_i^W M$ for every object $M$ and every integer $i$ are semisimple and $\mathrm{Hom}$'s are finite dimensional $\mathbb{Q}$-vector spaces. A pure functor between two mixed categories is an additive functor which sends every pure object (an object $M$ with at most one nonzero graded quotient w.r.t. weight filtration) to a pure object of the same weight.
A counterexample(?): Let $\mathcal{M}$ be the category of finite $\mathbb{Q}$-representations of the quiver: $\bullet\to\bullet$. Then $\mathcal{M}$ is a mixed category with the weight filtration of a representation $f:V_0\to V_1$ given by $W_{-1}=0\to 0 , W_0=0\to V_1, W_1 = V_0\stackrel{f}{\to}V_1$. It has two simple pure objects $\mathbb{Q}\to 0$ and $0\to\mathbb{Q}$ and only one other indecomposable non-pure object $X = \mathbb{Q}\stackrel{\mathrm{id}}{\to}\mathbb{Q}$. Then I think that there are pure endofunctors of $\mathcal{M}$ which act as the identity on pure objects and send $X$ to either of the simple pure objects and these functors are not exact. So I think purity is not sufficient for exactness.
