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It is known that every ribbon category with unimodality allows symmetrized$6j$-symbols with full tetrahedral symmetries while a spherical category doesnot in general. We give an explicit counterexample for this, namely thecategory $\mathcal{E}$. We define the mirror conjugate symmetry of $6j$-symbolsinstead and show that $6j$-symbols of any unitary spherical category can benormalized to have this property.  As an application, we discuss an exactly soluble model on a honeycomblattice. We prove that the Levin-Wen Hamiltonian is exactly soluble andhermitian on a unitary spherical category.

On symmetrization of 6j-symbols and Levin-Wen Hamiltonian