  Given an (irreducible) Mobius covariant net $\mathcal A$, we prove a
Bisognano-Wichmann theorem for its categorical extension $\mathscr
E^{\textrm{d}}$ 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-local
extension of $\mathcal A$ obtained by a $C^*$-Frobenius algebra $Q$ in
$\textrm{Rep}^{\textrm{d}}(\mathcal A)$. As an application, we discuss the
relation between the domains of modular operators and the preclosedness of
certain unbounded operators in $\mathscr E^{\textrm{d}}$.
