Necessary condition for the existence of subobject classifier

Question

I was reading the article on subobject classifiers on

https://ncatlab.org/nlab/show/subobject+classifier

Well, in Section 6 the author writes: " there is an easy condition ensuring a category with a terminal object can’t have a subobject classifier: if there are no nonidentity morphisms out of the terminal object".

I do not understand the formal meaning of the necessary condition. Can you help me, please?

Answer 1

I suspect you're thinking too hard about this.

Remember how the subobject classifier $\Omega$ is defined! It says that every mono in your category is a pullback of one ~special mono~ $\mathsf{True} : 1 \rightarrowtail \Omega$.

But if there aren't any monos out of $1$, then there's no ~special mono~ out of $1$ either, and a subobject classifier cannot exist!

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I hope this helps ^_^