Definable categories and T-motives

Making use of Freyd’s free abelian category on a preadditive category we show that if $T:D\to \mathcal{A}$ is a representation of a quiver $D$ in an abelian category $\mathcal{A}$ then there is an abelian category $$\mathcal{A}$ (T)$,afaithfulexactfunctor$F_T: $\mathcal{A}$ (T) \to $\mathcal{A}$$ and an induced representation $\tilde{T}:D\to \mathcal{A}(T)$ such that $F_T\tilde T= T$universally\mathrm{.}Wethencanshowthat$$\mathbb{T}$$-motives as well as Nori’s motives are given by a certain category of functors on definable categories.