Quotient description of the ordinary subobject classifier

Let $cmathcal{V}(cmathcal{H})$ be the context category of the Hilbert space $cmathcal{H}$, let $cmathcal{V}_f(cmathcal{H})$ be the same category with the group action forgotten, let $F$ be the associated functor, and let $\underline{G}$ denote the relevant group object. Write $\underline{\Omega}^{\mathcal{V}_f(\mathcal{H})}$ and $\underline{\Omega}^{\mathcal{V}(\mathcal{H})}$ for the corresponding subobject classifiers. Quotient subobject-classifier conjecture. The ordinary subobject classifier is isomorphic to the quotient

$$\underline{\Omega}^{\mathcal{V}_f(\mathcal{H})}\simeq F(\underline{\Omega}^{\mathcal{V}(\mathcal{H})})/\underline{G}.$$

This follows in the source from the preceding result and the identification of the two underlying context categories; the displayed assertion is therefore solved in the paper rather than left open.