Subobject classifiers in categories of smooth spaces

Question

In "Elementary Categories, Elementary Toposes" (chapter 23), by Mclarty, as well as "A Primer of Infintesimal Analysis" by Bell, the authors describe various axioms which a topos may satisfy in order to support synthetic differential geometry. In other words, they describe "toposes of smooth spaces". I am not sure if this appeal makes sense, but I am wondering if somebody could give me some intuition about what the subobject classifier $\Omega$ "looks like" in such a topos of smooth spaces. Are there cases where $\Omega$ looks similar to some familiar real manifold ? I am aware that there are various different toposes which can support the axioms of synthetic differential geometry, so perhaps my question is too vague. But I am finding it challenging to get to grips with the technical details of toposes of smooth spaces, and would be very grateful if somebody could help me get a mental picture of $\Omega$ within a category of smooth spaces. I do not know if this question makes sense, and whether $\Omega$ can be easily visualized as a space with a specific structure, but I thought it worth asking, since understanding more about $\Omega$ could help me within my early steps into synthetic differential geometry.