My question is exactly that of the title. Given a topos $T$, is there a natural sufficient condition for a functor $f$ from $T$ to another topos, $S$, to the subobject classifier? (By this I mean of course that not only that $f$ sends $\Omega_T$ to $\Omega_S$, but also that $f$ acts on morphisms in the appropriate way with respect to the subobject classifier's defining diagram.)
2026-03-30 05:10:08.1774847408
What functors preserve subobject classifiers?
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A logical functor preserves the topos structure; in particular, preserves power objects, and so maps $\Omega_T = \mathcal{P} 1_T$ to $\Omega_S = \mathcal{P} 1_S$.