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

For a given family $\{({\mathrm{q}}_i,{\mathrm{t}}_i,{\mathrm{p}}_{\mathrm{i}}){\}}_{i\in I}$ of adjoint triples between exact categories $\mathcal{C}$ or $\mathcal{D}$, we show that any cotorsion pair in $\mathcal{C}$ and $\mathcal{D}$ 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 $\mathrm{Add}(\mathcal{A},R-Mod)$ equipped with an exact structure $\mathcal{E}$. Under mild conditions on $\mathcal{A}$, we introduce the stalk functor at any object of $\mathcal{A}$, and subsequently, we investigate cotorsion pairs induced by stalk functors. Finally, we use them to present an intrinsic characterization of projective/injective objects in $(Add(\mathcal{A},R-Mod);\mathcal{E})$.