Transfer of homological objects in exact categories via adjoint triples. Applications to functor categories

For a given family \(qᵢ, tᵢ, pᵢ ) ∈ I of adjoint triples between exact categories C or D, we show that any cotorsion pair in C and D yield two canonical cotorsion pairs providing a concrete description of objects without using any injectives/projectives object hypothesis. We firstly apply this result for the evaluation functor on the functor category Add(A, R -Mod) equipped with an exact structure E. Under mild conditions on A, we introduce the stalk functor at any object of A, and subsequently, we investigate cotorsion pairs induced by stalk functors. Finally, we use them to present an intrinsic characterization of projective/injective objects in (Add(A, R-Mod); E).