Is there a (concrete) category of superstructures? The superstructure $V(X)$ over a set $X$ is usually defined as follows.


*

*$V_0(X)$

*$V_{i+1}(X) = V_i(X) \cup P(V_i(X))$

*$V(X) = ⋃_{i=0}^∞V_i(X)$


This defines a metafunction $V.$
Remark. To make sure the $V$ is well-behaved, we should only apply it to sets $X$ consisting entirely of urelements. Otherwise, the structure of $V(X)$ may depend on more than just the cardinality of $X$.
Now intuitively, the domain of $V$ is the class of all abstract sets, and the codomain of $V$ is the class of all superstructures. Thus I'd like to know, is there a more structuralist definition of $V$? In particular, is there a concrete category of "superstructures" $\mathrm{Sus}$ such that $V : \mathrm{Set} \rightarrow \mathrm{Sus}$ can be defined as the left-adjoint to the forgetful functor $U : \mathrm{Sus} \rightarrow \mathrm{Set}$?
The idea is that the objects of $\mathrm{Sus}$ should be ordered pairs $(S,\in')$ where $\in'$ acts like a membership relation within the universe $S$.
 A: Whenever you have an endofunctor $T$ on a category $\mathcal{C}$ that you want to view as a left adjoint, it is necessary and sufficient to find natural transformations $\eta : \mathrm{id} \Rightarrow T$ and $\mu : T T \Rightarrow T$ such that $(T, \eta, \mu)$ is a monad; indeed, in that case, there exist a category $\mathcal{D}$ and an adjunction $F \dashv U : \mathcal{D} \to \mathcal{C}$ such that $T = U F$.
It is clear how to make this "superstructure" construction into an endofunctor $T$ on $\mathbf{Set}$, because each step of the construction is itself functorial. There is also an evident natural transformation $\eta : \mathrm{id} \Rightarrow T$. So we just have to find a suitable $\mu : T T \Rightarrow T$. 
Unfortunately, the obvious thing only works if we replace $\mathscr{P}$ with $\mathscr{P}_{< \aleph_0}$: because then the rank of every element of $T T X$ will be finite (if we count elements of $X$ as having rank 0) and so can be "evaluated" to yield an element of $T X$. The category of $(T, \eta, \mu)$-algebras also admits a simpler description in this case: they are sets $A$ equipped with a map $\mathscr{P}_{< \aleph_0} A \to A$. You may think of this as the algebraic theory with one $n$-ary operation $\{ -, \ldots, - \}$ for each natural number $n$, subject to axioms of the form $\{ x, x \} = \{ x \}$, $\{ x, y \} = \{ y, x \}$, etc. The initial algebra is, of course, $V_\omega$, where structure map $\mathscr{P}_{< \aleph_0} V_\omega \to V_\omega$ is (literally!) the identity map.
The above can be generalised to $\mathscr{P}_{< \kappa}$ for any regular cardinal $\kappa$, but then one has to take $\kappa$-many steps in the transfinite construction of free algebras instead of just $\omega$-many.
