I'm going through the Goldblatt's "Topoi", in particular, through the arrow-based interpretation of first-order logic in a topos (in $\mathbf{Set}$ for now).
There, he defines the arrow $\exists_A : \mathcal{P}(A) \rightarrow 2 \simeq 2^A \rightarrow 2$, which maps any non-empty set into $1$ and the empty set into $0$, like this:
$$\exists_A(B) = \begin{cases} 1 &\text{if} &B \neq \emptyset \\ 0 &\text{if} &B = \emptyset \end{cases}$$
It follows that $\exists_A$ is the character of the set
$$\begin{align}C &= \{ B : B \neq \emptyset \} \\ &= \{ B : \text{for some } x \in A, x \in B. \}\end{align}$$
But then if $\in_A \hookrightarrow \mathcal{P}(A) \times A$ is the membership relation on $A$ (§4.7), i.e.
$$\in_A = \{ \langle B,x \rangle : B \subseteq A, \text{ and } x \in B \},$$
we see that applying the first projection $p_A(\langle B,x\rangle) = B$ from $\mathcal{P}(A) \times A$ to $\mathcal{P}(A)$ yields $p_A(\in_A) = C$.
Thus $\exists_A$ is the character of the image of the composite
$$ \in_A \hookrightarrow \mathcal{P}(A) \times A \overset{P_A}{\longrightarrow} \mathcal{P}(A).$$
As a side note, he derived the analogous $\forall_A : 2^A \rightarrow A$ as the character of the exponential adjoint of $\top_A \circ \pi_A$, where $\pi_A : 1 \times A \rightarrow A$ and $\top_A : A \rightarrow 2$ are the obvious ones.
I typically try to come up with my own approach before reading on, and the way I defined $\exists_A$ is (heavily inspired by $\forall_A$) $\lnot \circ \text{char} \ulcorner f \urcorner$, where $\ulcorner f \urcorner : 1 \rightarrow 2^A$ is the exponential adjoint of $f : 1 \times A \rightarrow 2, f = \bot \circ \pi_1$ (so $\ulcorner f \urcorner (0) = a \mapsto 0$). So, intuitively and blurring the line between subsets and their characters, $\ulcorner f \urcorner$ selects the (character of the) empty subset of $A$. So, $\ulcorner f \urcorner$'s character in turn maps the empty subset into 1 and any other subset into 0. Applying $\lnot$, we get what we want: mapping the empty subset into 0 and any other subset into 1.
Even after reading Goldblatt's version, I find mine more intuitive (assuming it's correct), although I wasn't able to quickly prove (or disprove) that these arrows are the same in arbitrary topoi. Is this a legitimate definition? Does it work better (or worse, or what else) than Goldblatt's?
My gut feel is that it's related to intuitionistic vs classical treatment of quantifiers — that is, AFAIK $\lnot \forall x. P \Rightarrow \exists x. \lnot P$ is a theorem of the classical logic but not of the intuitionistic one, but I haven't read that much into the first-order logic in general topoi to figure this out.