The Drinfel'd codouble constuction for monoidal Hom-Hopf algebra

Let $(H, β)$ be a monoidal Hom-Hopf algebra with the bijective antipode $S$, In this paper, we mainly construct the Drinfel'd codouble $T(H)=(H^{op}\otimes H^{*}, β\otimes β^{*-1})$ and $\widehat{T(H)}=( H^{*}\otimes H^{op}, β^{*-1}\otimes β)$ in the setting of monoidal Hom-Hopf algebras. Then we prove both $T(H)$ and $\widehat{T(H)}$ are coquasitriangular. Finally, we discuss the relation between Drinfel'd codouble and Heisenberg double in the setting of monoidal Hom-Hopf algebras, which is a generalization of the part results in \cite{L94}.

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