Intuition behind definition of a subobject classifier How do you go from the intuition behind the subobject classifier (the codomain of characteristic functions) to its actual definition?

I'm trying to understand what an elementary topos is and am using the definition from this paper. Specifically, I am using definition 1.5 on page 5 (pdf page 5).

A topos (or elementary topos) is a cartesian closed category with finite limits and a subobject classifier.

I'm trying to understand how to connect the intuition behind what a subobject classifier is with this definition on Wikipedia.
The intuitive sense that I get is that a subobject classifier is very, very similar to the notion of a truth value in logic. In the category $\mathrm{Set}$, the subsets of a fixed set $X$ are in 1:1 correspondence with indicator functions from $X$ to $\{0, 1\}$. This makes sense to me. In a very loose, intuitive sense, it seems like tweaking the choice of $\{0, 1\}$ and changing it to $\mathbb{N}$ or $\{ x : x \in \mathbb{R} \land 0 \le x \le 1 \}$ would give us multisets or fuzzy sets. That is all intuition though. Assuming that intuition is correct, I'm trying to see how it leads to the actual definition.
For example, here is the commutative diagram from Wikipedia. This diagram is a pullback. I know the definition of a pullback, but I must confess that I don't know what it buys us here.
$$
\require{AMScd}
\begin{CD}
U @>>> 1 \\
@V{j}VV @VVV \\
X @>{\chi_j}>> \Omega
\end{CD}
$$
Additionally, $j$ is a monomorphism, and $\chi_j$ is the only element of the homset $\textrm{hom}(X, \Omega)$ associated with the arrow $j$.
The thing that I really don't understand about this picture is what the implied arrow of the diagram $\chi_j \circ j$ is supposed to be.
Pushing on the intuitive analogy, it's an arrow from an arbitrary set $U$ to something resembling a truth value.
However, I think the fact that $\chi_j \circ j$ also factors as a sequence of arrows $U \to 1 \to \Omega$ implies that, in the category $\mathrm{Set}$ at least, $\chi_j \circ j$ is a constant function. However, I have no idea when it's constantly true or constantly false. If I had to guess, I would say it's probably when $X \subset U$, but that is pure speculation on my part.
 A: It's worth noting that the vertical arrow $1 \to \Omega$ is also part of the data of a subobject classifier and is usually called $\mathrm{true}$ (or $\mathrm{t}$ or something to that effect).
So the fact that $\chi_j \circ j = \mathrm{true} \circ \mathrm{!}$ (where $\mathrm{!}$ is the terminal map to $1$) means that $\chi_j$ is intuitively equal to $\mathrm{true}$ everywhere in the image of $j$.
The fact that the diagram in question is a pullback means intuitively that $j \colon U \to X$ is the largest subobject of $X$ where $\chi_j$ is constantly true, meaning that $\chi_j$ isn't true anywhere but $U$. If you have any morphism $f \colon Y \to X$ and $\chi_j$ is true everywhere in the image of $f$, then $f$ factors through $j$.
More precisely, we can unpack the definition of pullback a bit. Given any morphism $f \colon Y \to X$ such that $\chi_j \circ f = \mathrm{true} \circ \mathrm{!}$, we get a unique morphism $\bar{f} \colon Y \to U$ such that $j \circ \bar{f} = f$. The maps to the terminal object don't matter, since they're all unique.
In particular, for any monomorphism $i \colon V \to X$, if $\chi_j \circ i = \mathrm{true} \circ \mathrm{!}$, then $V$ is a subobject of $U$ via $\bar{i} \colon V \to U$. The fact that $\bar{i}$ is a monomorphism follows from the equation $j \circ \bar{i} = i$ and the hypothesis that $i$ is a monomorphism. So what this means is that $U$ contains any subobject that $\chi_j$ is constantly true on.


and $\chi_j$ is the sole element of the homset $\textrm{hom}(X,\Omega)$.

Not exactly. The definition of $\Omega$ means that maps $X \to \Omega$ are in bijection with subobjects of $X$. You might have been confused by phrases like

there is a unique map $\chi \colon X \to \Omega$ such that...

But $\chi$ here isn't the only map; it's the only map such that a certain square is a pullback. In other words, $\chi_j$ is the only map from $X$ to $\Omega$ that makes $j \colon U \to X$ the largest subobject of $X$ where $\chi_j$ is constantly true.
