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Given an (irreducible) Mobius covariant net $\mathcal A$, we prove aBisognano-Wichmann theorem for its categorical extension $\mathscrE^{\textrm{d}}$ associated to the braided $C^*$-tensor category$\textrm{Rep}^{\textrm{d}}(\mathcal A)$ of dualizable (more precisely"dualized") Mobius covariant $\mathcal A$-modules. As a closely related result,we prove a (modified) Bisognano-Wichmann theorem for any (possibly) non-localextension of $\mathcal A$ obtained by a $C^*$-Frobenius algebra $Q$ in$\textrm{Rep}^{\textrm{d}}(\mathcal A)$. As an application, we discuss therelation between the domains of modular operators and the preclosedness ofcertain unbounded operators in $\mathscr E^{\textrm{d}}$.

Bisognano-Wichmann property for rigid categorical extensions and non-local extensions of conformal nets