We define the right regular dual of an object X in a monoidal category C , and give several results regarding the weak rigid monoidal category. Based on the definition of the right regular dual, we construct a weak Hopf algebra structure of H = End(F) whenever (F, J) is a fiber functor from category C to V ec and every X ∈ C has a right regular dual. To conclude, we give a weak reconstruction theorem for a kind of weak Hopf algebra.
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