Two Category Theory Problems I have two problems and I need some help or ideas about how to solve them.

Suppose I have the following Category:
  
  
*
  
*the objects are structures $(X, Cn)$
  
  
  i) X is any set
ii) $Cn$ is a map from the power set of X to the power
  set of X, i.e. $Cn : \mathscr{P}(X) \to \mathscr{P}(X)$
  
  
*
  
*The arrows are defined as follows:
  
  
  Given $(X, Cn)$ and $(X', Cn')$, an arrow $t$ goes from $X$ to $X'$, and is
  injective, such that for all $A \subset X$, $$t(Cn(A))=Cn'(t(A)).$$
Now, consider the following definition: Given $(X, Cn)$, and $A \subset X$, $$Cnp(A)= U\{Cn(A')/A' \subset A,\ \text{and} Cn(A')≠ X\}$$
We define the functor F as:
i) $F(X, Cn) = (X, Cnp)$
ii) $F(t) = t$.

My problems are:
1) Is this functor idempotent, that is, $F(F(Cn))=F(Cn)$ and $F(F(t))=F(t)$?  
2) How can I define product in these categories?
Thanks!
 A: Let $X$ be a set with at least $2$ elements and call one of those two elements $a$.  Define
$$Cn(S) = \begin{cases} X & \text{if} \ |S| \geq 2 \\ X \setminus S & \text{if} \ |S| = 1 \\ \{a\} & \text{if} \ |S| = 0\end{cases}$$
where $|S|$ means the cardinality of $S$.  Now $F(X, Cn) = (X, Cnp)$ where
$$Cnp(S) = \begin{cases} X & \text{if} \ |S| \geq 2 \ \text{or} \ S = \{a\} \\ X \setminus S & \text{if} \ |S| = 1 \ \text{and} \ S \neq \{a\} \\ \{a\} & \text{if} \ |S| = 0\end{cases}$$
Finally look at $F(X, Cnp)$.  You should be able to see that the function on power sets sends $\{a\}$ to itself, so $F(X, Cnp) \neq (X, Cnp)$, meaning $F^2(X, Cn) \neq F(X, Cn)$.
As for products, one thing to notice is that your maps in this category are all injections $X \hookrightarrow Y$ with the property that the power set map on $X$ is basically a restriction of the power set map on $Y$.  So for any collection of objects $(X_i, Cn_i)$ if the product $\prod_i(X_i, Cn_i)$ exists then it comes with maps $\prod_i(X_i, Cn_i) \to (X_j, Cn_j)$ for each $j$ and so the product can be thought of as a subobject of each $(X_j, Cn_j)$.  I suggest you pick an $(X_1, Cn_1)$ and define the product $\prod_i(X_i, Cn_i)$ to be a subobject of $(X_1, Cn_1)$.  Maybe take the union of all injections $Z \hookrightarrow X_1$ that occur as part of a family of maps $\phi_i(Z, Cn) \to (X_i, Cn_i)$ for some $Cn$.
