= Subobject classifier of a coalgebra topos
{title2=$\Omega_G=\operatorname{Eq}(1,G\kappa\,\delta_\Omega)$}
Let $\kappa:G\Omega\to\Omega$ classify $G\top:1\hookrightarrow G\Omega$. In the <cofree coalgebra> $R\Omega$, equalize the identity and $G\kappa\,\delta_\Omega$. A map $\chi:X\to\Omega$ classifies a subcoalgebra exactly when $\chi=\kappa G\chi\,x$, since this says the subobject equals the inverse image of its image under $G$. Transposition gives precisely the displayed equalizer condition.
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