CatDat

Implication Details

Assumptions: subobject classifier, thin

Conclusions: trivial

Reason: Let $\mathcal{C}$ be a thin category with a subobject classifier $\Omega$. Every object has a subobject, hence a morphism to $\Omega$. It must be unique since $\mathcal{C}$ is thin. Hence, $\Omega$ is terminal. If $X$ is any object, then $1 \cong \mathrm{Hom}(X,\Omega) \cong \mathrm{Sub}(X)$ shows that every morphism $Y \to X$ is isomorphic to $\mathrm{id}_X : X \to X$. Apply this to $X = \Omega$ to finish the proof.