The aim of this paper is to construct a new braided T-category via the generalized Yetter-Drinfel'd modules and Drinfel'd codouble over a Hopf algebra, an approach different from that proposed by Panaite and Staic [1]. Moreover, in the case of finite dimensional, we will show that this category coincides with the corepresentation of a certain coquasitriangular Turaev group algebra which we construct. Finally we apply our theory to the case of group algebra.