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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 thecategory $\operatorname{Fun}(\Sigma,\mathcal{A})$ 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 thispaper 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 anapplication of our findings, we will give new proofs of known results on theexactness of limits and colimits in abelian categories.

Exactness of limits and colimits in abelian categories revisited