We introduce the notion of a lazy $2$-cocycle over a monoidal Hom-Hopf algebra and determine all lazy $2$-cocycles for a class of monoidal Hom-Hopf algebras. We also study the extension of lazy $2$-cocycles to a Radford Hom-biproduct.

In this paper, we mainly provide a categorical view on the braided structures appearing in the Hom-quantum groups. Let be a monoidal category on which F is a bimonad, G is a bicomonad, and ϕ is a distributive law, we discuss the necessary and sufficient conditions for , the category of mixed bimodules to be monoidal and braided. As applications, we discuss the Hom-type (co)quasitriangular structures, the Hom–Yetter–Drinfeld modules, and the Hom–Long dimodules.

Let L,α be a Hom-Lie-Yamaguti superalgebra. We first introduce the representation and cohomology theory of Hom-Lie-Yamaguti superalgebras. Also, we introduce the notions of generalized derivations and representations of L,α and present some properties. Finally, we investigate the deformations of L,α by choosing some suitable cohomology.

In this paper, we first introduce the notion of Hom–Leibniz bialgebras, which is equivalent to matched pairs of Hom–Leibniz algebras and Manin triples of Hom–Leibniz algebras. Additionally, we extend the notion of relative Rota–Baxter operators to Hom–Leibniz algebras and prove that there is a Hom–pre-Leibniz algebra structure on Hom–Leibniz algebras that have a relative Rota–Baxter operator. Finally, we study the classical Hom–Leibniz Yang–Baxter equation on Hom–Leibniz algebras and present its connection with the relative Rota–Baxter operator.