Implication Details

Claim: If a category has a regular subobject classifier and has a strict terminal object, then it is thin.

Proof: Let $\Omega$ be a regular subobject classifier. Since $1$ is a strict terminal object, $\top : 1 \to \Omega$ is an isomorphism. This implies that every regular monomorphism is an isomorphism. Hence, by taking the equalizer of two parallel morphisms, we see that the category is thin.

This implication has a dual.