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In this paper, we define and study weak monoidal Hom-Hopf algebras, whichgeneralize both weak Hopf algebras and monoidal Hom-Hopf algebras. If $H$ is aweak 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 thecategories ${_{H}\mathcal{WMHYD}^{H}}(\alpha,\beta)$ as components.

Constructing New Braided $T$-Categories via Weak Monoidal Hom-Hopf Algebras