Abstract: Firstly, by providing  a Hom-Lie algebra and its representation,  we proved that a strong quasi-trace function on  a Hom-Lie algebra could induce a 3-Hom-Lie algebra and its representation, thereby proving that a generalized Reynolds operator on a Hom-Lie algebra was also  a generalized Reynolds operator on the  induced 3-Hom-Lie algebra. Secondly, we studied the mutual derivation properties of Hom-NS-Lie algebras and generalized Reynolds operators, and gave the adjoint relation of the corresponding categories.

Key words: Hom-Lie algebra, generalized Reynolds operator, 3-Hom-Lie algebra, Hom-NS-Lie algebra