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

For a given family $\{(\mathrm{q}_i, \mathrm{t}_i, \mathrm{p_i} )\}_{i \in I}$ 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 $\operatorname{Add}(\mathcal{A}, R \mbox{-Mod})$ 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 $(\mbox{Add}(\mathcal{A}, R\mbox{-Mod}); \mathcal{E})$.