  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
$(\mbox{Add}(\mathcal{A}, R\mbox{-Mod}); \mathcal{E})$.
