On occasion I've informally written statements like:
"For any set $X$ we let $\mathcal{L}(X)=X\times X\cup \{X,(\emptyset,X)\}$"
Now this has never really been a problem because most of the time, I could with additional effort find a suitable family of sets $\mathcal{F}$ for which every set under consideration in my thoughts was a member of. Thus if I wished I could have gone and written out this more formally as:
"Let $\mathcal{L}$ be a function with $\text{dom}(\mathcal{L})=\mathcal{F}$ defined for any $X\in \mathcal{F}$ by $\mathcal{L}(X)=X\times X\cup \{X,(\emptyset,X)\}$"
Yet I'm unsure how to express this same function formalization when dealing with proper classes.
How would one symbolize what I can best describe as an "endofunction on the class of all sets"
Additionally can one consider proper classes or some variation of them as algebraic structures? For example something like the join-semi lattice of all sets, consisting of the union operator acting on the class of all sets. (Again I'm not sure of the exact names of the mathematical objects I'm trying to describe here so if I have referred to something with an informal/improper name I apologies).
We use formulas: an appropriate formula in $n+1$ free variables represents a class function in $n$ variables. Specifically, suppose $\varphi(x_1, ..., x_n, y)$ is a formula with only the displayed free variables, such that for every $x_1, ..., x_n$ there is exactly one $y$ such that $\varphi(x_1, ..., x_n, y)$ holds. Then $\varphi$ defines a class function $F_\varphi$ sending $(x_1, ..., x_n)$ to the unique $y$ making $\varphi$ true.
For example, the class function "$S: x\mapsto \{x\}$" can be represented by the formula $$\varphi_S(x, y)\equiv \forall z(z\in y\iff z=x).$$ Your class function is a bit more complicated, but still not too bad.
We can then use the defining formula in place of the function we have in mind. For instance, if we want to say "For all $x$, $S(x)$ has property $P$," we can say this as "For all $x$ and all $y$, if $\varphi_S(x, y)$ then $y$ has property $P$." So we never actually need to refer to the class function itself, we can always use the formula defining it instead. In this view, a statement like "For any set $X$ we let $\mathcal{L}(X)=$[stuff]" is thought of as a more natural way to introduce the corresponding formula in two free variables.