Symmetries and the u-condition in Hom-Yetter-Drinfeld categories

Let (H, S, α) be a monoidal Hom-Hopf algebra and [Formula: see text] the Hom-Yetter-Drinfeld category over (H, α). Then in this paper, we first find sufficient and necessary conditions for [Formula: see text] to be symmetric and pseudosymmetric, respectively. Second, we study the u-condition in [Formula: see text] and show that the Hom-Yetter-Drinfeld module (H, adjoint, Δ, α) (resp., (H, m, coadjoint, α)) satisfies the u-condition if and only if S² = id. Finally, we prove that [Formula: see text] over a triangular (resp., cotriangular) Hom-Hopf algebra contains a rich symmetric subcategory.