Consider $\presheaves{\mathbf{P}}$ where $\mathbf{P} = (P, \leq)$ is a poset considered as a category. We wish to characterize the subobject classifier of $\presheaves{\mathbf{P}}$ in this special case. A subset $D \subset P$ is a downset if $a \leq b \in D$ implies $a \in D$. We use $\downset p = \setbuilder{x}{x \leq p}$ to denote the downset generated by $p$, and $\mathcal{D}(S)$ to denote the set of downsets of $S \subseteq P$. A sieve on $p \in \mathbf{P}$ is then precisely an element of $\mathcal{D}(\downset p)$. Thus, the subobject classifier can be defined with $\Omega(p) = \mathcal{D}(\downset p)$ and $\Omega(p \leq q) : \Omega(q) \rightarrow \Omega(p)$ the map that sends $D$ to $\downset p$ if $\downset p \subseteq D$ or itself otherwise.

The terminal presheaf is still $\Delta T$, taking constant value $\set{T}$. The monic $\true : \Delta T \hookrightarrow \Omega$ will have components $\true_p : \set{T} \rightarrow \Omega(p)$ sending $T$ to $\downset p$ (the maximal sieve). Take a monic $G \hookrightarrow F$, which we may consider to be an actual subfunctor in the sense that $G(p) \subseteq F(p)$ for all $p$ and $G(p \leq q)$ is a restriction of $F(p \leq q)$ for all arrows. The characteristic function $\chi : F \rightarrow \Omega$ has components $\chi_p : F(p) \rightarrow \Omega(p)$ with $\chi_p(\alpha) = \setbuilder{x}{x \leq p \tand F(x \leq p)(\alpha) \in G(x)}$.

Consider $\mathbf{Q} = (\mathbb{Q}, \leq)$ as a category and consider $\presheaves{\mathbf{Q}}$. In this case, a downset of rational numbers corresponds to a one-sided Dedekind cut, thus each corresponds precisely to a real number via its supremum (see here for a discussion). However, we also consider $\varnothing$ a downset appearing in every $\Omega(p)$; we can represent this as an adjoined value $-\infty$. From this point of view, $\presheaves{\mathbf{Q}}$ has a subobject classifer $\Omega$ with $\Omega(p) = \setbuilder{r \in \mathbb{R} \cup \set{-\infty}}{r \leq p}$.