  Let $H_1$ and $H_2$ be Hopf algebras which are not necessarily finite
dimensional and $\alpha,\beta \in Aut_{Hopf}(H_1), \gamma,\delta \in
Aut_{Hopf}(H_2)$. In this paper, we introduce a category
${}_{H_1}\mathcal{LR}_{H_2}(\alpha, \beta, \gamma, \delta)$, generalizing
Yetter-Drinfeld-Long bimodules and construct a braided $T$-category
$\mathcal{LR}(H_1,H_2)$ containing all the categories
$_{H_1}\mathcal{LR}_{H_2}(\alpha, \beta, \gamma, \delta)$ as components. We
also prove that if $(\alpha, \beta, \gamma, \delta)$ admits a quadruple in
involution, then ${}_{H_1}\mathcal{LR}_{H_2}(\alpha, \beta, \gamma, \delta)$ is
isomorphic to the usual category ${}_{H_1}\mathcal{LR}_{H_2}$ of
Yetter-Drinfeld-Long bimodules.
