The subobject classifier is a set?? I'm reading about subobject classifiers in presheaf categories, and got confused.
The book speaks of a subobject classifier $\Omega$ of the presheaf category $[P^{op}, Set]$, where $P$ is a poset having a largest element $T$. So, as I understand, $\Omega$ is an element of $[P^{op}, Set]$, meaning it is a contravariant functor from $P$ to $Set$.
But then a few lines later, the book says we can regard the truth values of the subobject classifier as a Heyting Algebra, and this induces a monoid structure on $\Omega$. The book goes on to speak of actions of this monoid on a set, all the while using notation like $\alpha \in \Omega$. In other words, it looks like they are saying $\Omega$ as a set.
My question is, how can $\Omega$ have any elements, if it is a functor? Are they writing $\Omega$ when they actually mean $\Omega$ applied to $T$? If this is the case then my confusion is (mostly) cleared.
 A: If by "element" they mean "global element", then a global element (of $A$) is an arrow from $1\to A$ where $1$ is the terminal object. Since $T$ is the top element of the poset, it is the terminal object of that poset when viewed as a category. Since $\mathsf{Hom}$ is continuous in its second argument, we have that $\mathsf{Hom}(-,T)\cong 1$ where $1$ is the terminal object of the category of presheaves. A global element of the presheaf $P$ is an element of $$\mathsf{Nat}(1,P)\cong\mathsf{Nat}(\mathsf{Hom}(-,T),P)\cong P(T)$$ where $\mathsf{Nat}(F,G)$ is the set of natural transformations between $F$ and $G$, i.e. the hom-set of the category of presheaves. The second isomorphism is Yoneda. So the global elements of the presheaf $\Omega$ are in correspondence with the elements of the set $\Omega(T)$.
More naturally, we can use the notion of the internal language of an elementary topos which can be presented as the Mitchell-Bénabou language. This lets us write what looks like "normal" mathematics but interpret it into any elementary topos. In this context, global elements correspond to closed terms. In other words, the elements of $\Omega(T)$, in this case, correspond to the closed terms of type $\Omega$ in the internal language.
