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Let $H$ be a finite-dimensional bialgebra. In this paper, we prove that the category $\mathrm{ℒℛ}\left(H\right)$ of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category ${}_{H\otimes H^{*}}^{H\otimes H^{*}}\mathrm{𝒴𝒟}$ over the tensor product bialgebra $H\otimes H^{*}$ as monoidal categories. Moreover if $H$ is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.

In this paper, we study the $H^{\sigma -R}$ type Hopf algebras and present its braided and quasitriangular Hopf algebra structure. This generalizes well-known results on $H^{\sigma }$ and $H^R$ type Hopf algebras. Finally, the classification of $H^{\sigma -R}$ type Hopf algebras is given.