  In this paper, we define and study weak monoidal Hom-Hopf algebras, which
generalize both weak Hopf algebras and monoidal Hom-Hopf algebras. If $H$ is a
weak monoidal Hom-Hopf algebra with bijective antipode and let $Aut_{wmHH}(H)$
be the set of all automorphisms of $H$. Then we introduce a category
${_{H}\mathcal{WMHYD}^{H}}(\alpha,\beta)$ with $\alpha,\beta\in Aut_{wmHH}(H)$
and construct a braided $T$-category $\mathcal{WMHYD}(H)$ that having all the
categories ${_{H}\mathcal{WMHYD}^{H}}(\alpha,\beta)$ as components.
