In local cohomology the ideal transform functors with respect to a pair of ideals are defined by $D_{I,J}(-)=\underset{\textbf{a} \in \tilde{w} (I,J)} {\varinjlim}\,\,\text D_{\textbf{a}}(-)$. Recall that
$\tilde{w} (I,J):=\lbrace \textbf{a}: \textbf{a} \text{is an ideal of R}, \exists n \in \Bbb{N} , I^n \subseteq \textbf{a}+J\rbrace$. Now I have a question. If $D_{I,J}(-)$ is an exact functor are all of $D_{\textbf{a}}(-)$ exact functors? If $D_{\textbf{a}}(-)$ is not an exact functor for an ideal $\textbf{a}\in\tilde{w} (I,J)$ then is not $D_{I,J}(-)$ an exact functor too?