In this paper, we get some identities between extension functors of local cohomology modules. Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$, $M$ an $R$-module and $t$ an integer. Let $N$ be a finitely generated $I$-torsion $R$-module. It is shown that $\Hom_R(N, H^{t}_{I,J}(M))\cong\Ext_R^{t}(N, M)\cong\Hom_R(N, H^{t}_{I}(M))\cong H^{t}_{I}(N,M)$, whenever $H^{i}_{I,J}(M)=0$ for all $i<t$.
