Purity and flatness in symmetric monoidal closed exact categories

Let (𝒜,−⊗−) be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. When 𝒜 is endowed with an injective cogenerator with respect to the exact structure, we show that an object ℱ in 𝒜 is flat if and only if any conflation ending in ℱ is pure. Furthermore, we prove a generalization of the Lambek Theorem (J. Lambek, A module is flat if and only if its character module is injective, Canad. Math. Bull. 7 (1964) 237–243) in 𝒜. In the case 𝒜 is a quasi-abelian category, we prove that 𝒜 has enough pure injective objects.