CatDat

Implication Details

Assumptions: regular subobject classifier, strict terminal object

Conclusions: thin

Reason: 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.