Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$, $M$ an $R$-module and $t$ an integer. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $N$ be a finitely generated $R$-module with $\Supp_R N={\rm V}(\frak{a})$ for some $\frak{a}\in \tilde{{\rm W}}(I,J)$. It is shown that if $\Ext_R^{j}(N, H^{i}_{I,J}(M))\in S$ for all $i
