  Let $ Aut_{mHH}(H)$ denote a set of all automorphisms of a monoidal Hopf
algebra $H$ with bijective antipode in the sense of Caenepeel S. and Goyvaerts
I. (Commun. Algebra 39, 2216-2240, 2011) and let $G$ be a crossed product group
$ Aut_{mHH}(H)\times Aut_{mHH}(H)$. The main aim of this paper is to provide
further examples of braided $T$-category in the sense of Turaev (1994, 2008).
For this purpose, we first introduce a class of new categories $_{H}\mathcal
{MHYD}^{H}(A, B)$ of monoidal Hom $(A, B)$-Yetter-Drinfeld modules with $A, B
\in Aut_{mHH}(H)$. Then we show that the category ${\cal
MHYD}(H)=\{{}_{H}\mathcal {MHYD}^{H}(A, B)\}_{(A, B)\in G}$ forms a braided
$T$-category, generalizing the main constructions construction by Panaite and
Staic (Isr J Math 158:349-365, 2007).
