 Subobject classifier of a coalgebra topos

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.