New Braided $T$-Categories over Hopf (co)quasigroups

Let $H$ be a Hopf quasigroup with bijective antipode and let $Au{t}_{HQG}(H)$ be the set of all Hopf quasigroup automorphisms of $H$. We introduce a category ${}_H{\mathcal{Y}\mathcal{D}\mathcal{Q}}^H(\alpha ,\beta )$ with $\alpha ,\beta \in Au{t}_{HQG}(H)$ and construct a braided $T$-category $\mathcal{Y}\mathcal{D}\mathcal{Q}(H)$ having all the categories ${}_H{\mathcal{Y}\mathcal{D}\mathcal{Q}}^H(\alpha ,\beta )$ as components.