Suppose that R is a commutative Noetherian ring with identity, I, J are ideals of R, and let M be a finitely generated R-module. Let HiI,J(−) be the ith local cohomology functor with respect to (I,J). In this paper, we show that the R-module HomR(R/I,H1I,J(M)/JH1I,J(M)) is always finitely generated. Moreover, we provide sufficient conditions such that the modules HomR(R/I,HiI,J(M)/JHiI,J(M))orTorRj(R/I,HiI,J(M)/JHiI,J(M)) is finitely generated.

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