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Let(C, α) and(H, β) be Hom-bialgebras and ω : C H→ H C a linear map. We introduce a Hom-ω-smash coproduct(Cω■H, γ) and give necessary and sufficient conditions for(Cω■H, γ) to be a Hom-bialgebra. We study the quasi-triangular structures over(Cω■H, γ)and