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