ZFC with $\in^*$ as a category -- What category is it? I was thinking earlier today about $\in^*$, the transitive and reflexive closure of $\in$. Then I realized that it forms a category when the objects are taken to be the sets of ZFC. I'm curious what this category is called or if it's an instance of a well-known flavor of category. It seems to have some interesting properties related to arrow factorization.

Let $A \in B \in C$ be an orthographic abbreviation for $A \in B \land B \in C$ and likewise for $\in^*$.
Let upper Latin letter be free variables. Let lower Latin vowels be variables bound by a universal quantifier and lower Latin consonants by variables bound by any other quantifier.
$\in$ is definable given $\in^*$, in particular, the following is true.
$$ A \in B \iff A \ne B \land (\forall e \mathop. (A \in^* e \in^* B) \to (A = e \lor e = B)) $$
Then I started thinking that the class of sets equipped with $\in^*$ looks like a poset category. If we say that $\text{hom}(A, B)$ has a unique arrow if and only if $A \in^* B$ and is otherwise empty, then we have a poset category.
This category has at least one interesting property: every arrow is the product of a finite number of prime arrows. However, the arrows don't factorize uniquely and the number of arrows in different decompositions might be different.
This category also has some unremarkable properties, such as an initial element $\varnothing$.
 A: As you pointed out yourself in the question, what you've defined is a partial order on $V$. It has a level for every ordinal, and the elements at level $\alpha$ are exactly the elements of $V_{\alpha+1}\setminus V_\alpha$, i.e., the elements of rank $\alpha$. Every element is above the elements in its transitive closure. I'm not sure what else there is to say about it.
The definition of $\in$ from $\in^*$ that you wrote down does not work. For example, $\emptyset\in \{\emptyset,\{\emptyset\}\}$, but the relation $\emptyset\in^* \{\emptyset,\{\emptyset\}\}$ can be non-trivially factored as $\emptyset \in^* \{\emptyset\}\in^* \{\emptyset,\{\emptyset\}\}$.
It's possible that an alternative construction is closer to what you had in mind. Define a category with objects $V$, where an arrow $A\to B$ is a finite sequence of sets $C_0,\dots,C_n$ with $A = C_0$, $B = C_n$, and $C_i\in C_{i+1}$ for all $i$. There's an obvious associative composition of sequences, and the identity arrow $A\to A$ is the sequence $A$ of length $1$. So we get a category, but now there can be multiple arrows between elements. For example, there are two arrows $\emptyset\to \{\emptyset,\{\emptyset\}\}$: (1) $\emptyset\in \{\emptyset,\{\emptyset\}\}$ and (2) $\emptyset\in \{\emptyset\}\in \{\emptyset,\{\emptyset\}\}$.
This is an instance of the path category construction: it is the free category generated by the directed graph $(V,\in)$.
In this category, an instance of $\in$ is an irreducible arrow, as you wanted. But now $\emptyset$ is no longer initial (it is only weakly initial): for every set $X$ other than $\emptyset$ and $\{\emptyset\}$, there is more than one arrow $\emptyset\to X$.
