= Hyperconnected-localic factorization
{title2=$\mathcal O(L)=f_*\Omega_{\mathcal E}$}
For a <geometric morphism> $f:\mathcal E\to\mathcal F$, the displayed internal <frame> represents $X\mapsto\operatorname{Sub}_{\mathcal E}(f^*X)$. The localic part is $\operatorname{Sh}_{\mathcal F}(L)\to\mathcal F$. Its basic open families pull back to the subobjects of $f^*X$ they represent; gluing these gives a full faithful, subobject-closed inverse image, hence the <hyperconnected geometric morphism> part. The represented subobject functor recovers the internal frame, proving uniqueness up to equivalence over $\mathcal F$.
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