Categories admitting singleton-classifiers and characterization of the category $\mathbf{Set}$ Trying to characterize categories equivalent to the category of sets, I have discovered (for myself) that instead of requiring that the coprojection morphism $\mathsf{true}:1\to \Omega=1\sqcup 1$ is a subobject classifier, it suffices to require that this morphism is an singleton classifier, which means that for every morphism $x:1\to X$ there exists a morphism $\chi_x:X\to \Omega$ such that  for any morphism $y:1\to X$ the equality $\chi_x\circ y=\mathsf{true}$ is equivalent to $x=y$.

Question. Is the notion of a singleton classifier essentially weaker than that of subobject classifier? Has it been already considered in the literature and if yes, under which terminology?

Using element classifiers I can prove the following characterizations in von Neumann-Bernays-Godel axiomatic system:

Theorem 1. A category $\mathcal C$ is equivalent to the category of sets if and only if $\mathcal C$ has the following properties:
1) $\mathcal C$ is locally small;
2) $\mathcal C$ is balanced (mono+epi = iso);
3) $\mathcal C$ has a terminal object $\mathtt 1$;
4) $\mathtt 1$ is a $\mathcal C$-generator;
5) $\mathcal C$ has equalizers;
6) $\mathcal C$ has arbitrary coproducts;
7) $|\mathsf{Mor}(\mathtt 1,\mathtt 1\sqcup \mathtt 1)|=2$;
8) the morphism $\mathsf{true}:\mathtt 1\to \mathtt 1\sqcup\mathtt 1$ is an singleton classifier in $\mathcal C$.

$\phantom{m}$

Theorem 2. A category $\mathcal C$ is isomorphic to the category of sets if and only if $\mathcal C$ has the following properties:
$(1)-(8)$ from Theorem 1;
(9) $\mathcal C$ has a unique initial object;
(10) for any noninitial $\mathcal C$-object $x$ the class of $\mathcal C$-objects that are isomorphic to $x$ is a proper class.

I have a feeling that these characterizations are known. If yes, to whom should they be attributed?
Added in Edit. I have found something quite close to the above characterizations in nLab.
 A: I don't think this has been considered. Mainly I've never seen it, but also there are specific feature of this notions that makes it unlikely to be a relevant category theoretic notion independently of your other conditions:


*

*It is not really a universal property, in the sense that it does not characterize what are morphisms to $\Omega$ as not all morphisms to $\Omega$ classify a singleton.

*Your uniqueness conditions only involves the behavior of the map $X \to \Omega$ on elements of $X$. This is of course very natural in your situation as all map in the category of sets are determined by their value on elements (and this is implied by your axiom that $1$ is a generator) but this is a very weird condition in category where $1$ is not assumed to be a generator. For example, I don't think a subobject classifier will be an element classifier in general.
Regarding an example where  this is different from a sub-object classifier:
If I have a model $M$ of IZF or CZF, then in the category $S$ of sets of $M$, $2=1 \coprod 1$ classifies only complemented subobjects. (In a model of IZF there would be an actual subobject classifier $\Omega$, with $2 \subset \Omega$, but not necessarily in a model of CZF).
If I restrict to the full subcategory $D \subset S$ of objects that are decidable (i.e. the set $X$ such that the diagonal inclusion $X \to X \times X$ is complemented) It is a classical category theoretic fact that $D$ is stable under finite limits (because $S$ being an extensive category decidable will be stable under finite product, and a sub-object of a decidable object is decidable).
Now, all singleton in $D$ are complemented as a singleton $a:1 \to X$ can be written as pullback of $1 \times X \to X \times X$ along $X \to X\times X$, so $2= 1 \coprod 1$ will indeed be an 'element classifier'. Explicitely, as $X$ is decidable, there is a map $\delta:X \times X \to 2$ that classifies the diagonal, and given $x: 1 \to X$, $\{x\}$ is classified by $\delta(x, \_ )$.
However it does not classify all subobjects (unless the law of excluded middle holds in $M$ of course) for example as soon as there are a non-complemented subset of $\mathbb{N}$ in $M$, then as $\mathbb{N}$ is always a decidable object that will give you a subobject not classified by $2$.
Note: I initially tried to use a sheaf model to get a more explicit category, but I got into trouble due to the fact that in a sheaf model $1$ is almost never a generator. And as I mentioned at the begining, your definition is a bit unnatural if we do not assume that $1$ is a generator. I realized I do not know how to construct well-pointed toposes not satisfying LEM, other than going through the kind of filter-quotient constructions that produces models of IZF from sheaves model...
Note 2: What do you think of the name "singleton classifier" instead of "element classifier" ?
