  Let $\Sigma$ be a small category and $\mathcal{A}$ be a $\Sigma$-co-complete
(resp. $\Sigma$-complete) abelian category. It is a well-known fact that the
category $\operatorname{Fun}(\Sigma,\mathcal{A})$ of functors of $\Sigma$ in
$\mathcal{A}$ is an abelian category, and that the functor
$\mathsf{colim}_\Sigma(-):\operatorname{Fun}(\Sigma,\mathcal{A})\rightarrow\mathcal{A}$
(resp.
$\mathsf{lim}_{\Sigma}(-):\operatorname{Fun}(\Sigma,\mathcal{A})\rightarrow\mathcal{A}$)
is left (resp. right) adjoint to
$\kappa^{\Sigma}:\mathcal{A}\rightarrow\operatorname{Fun}(\Sigma,\mathcal{A})$,
where $\kappa^{\Sigma}$ is the associated constant diagram functor. In this
paper we will show that the functor $\mathsf{colim}_\Sigma(-)$ (resp.
$\mathsf{lim}_{\Sigma}(-)$) is exact if and only if the pair of functors
$\left(\mathsf{colim}_\Sigma(-),\kappa^{\Sigma}\right)$ (resp.
$\left(\kappa^{\Sigma},\mathsf{lim}_{\Sigma}(-)\right)$) is Ext-adjoint. As an
application of our findings, we will give new proofs of known results on the
exactness of limits and colimits in abelian categories.
