Exactness of limits and colimits in abelian categories revisited
Let $Σ$ be a small category and $\mathcal{A}$ be a $Σ$-co-complete (resp. $Σ$-complete) abelian category. It is a well-known fact that the category $\operatorname{Fun}(Σ,\mathcal{A})$ of functors of $Σ$ in $\mathcal{A}$ is an abelian category, and that the functor $\mathsf{colim}_Σ(-):\operatorname{Fun}(Σ,\mathcal{A})\rightarrow\mathcal{A}$ (resp. $\mathsf{lim}_Σ(-):\operatorname{Fun}(Σ,\mathcal{A})\rightarrow\mathcal{A}$) is left (resp. right) adjoint to $κ^Σ:\mathcal{A}\rightarrow\operatorname{Fun}(Σ,\mathcal{A})$, where $κ^Σ$ is the associated constant diagram functor. In this paper we will show that the functor $\mathsf{colim}_Σ(-)$ (resp. $\mathsf{lim}_Σ(-)$) is exact if and only if the pair of functors $\left(\mathsf{colim}_Σ(-),κ^Σ\right)$ (resp. $\left(κ^Σ,\mathsf{lim}_Σ(-)\right)$) 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.