I'm trying to get started learning category theory. A problem I'm working on is to show that for a set $S$, the partial order $(\mathcal{P}(S),\subseteq)$ viewed as a category is cartesian closed.
So far, I was thinking that $S$ is the terminal object, and that the product of any two subsets $A \times B$ always exists (the greatest lower bound or intersection of $A$ and $B$), but am having trouble completing the proof by showing that the category has exponentiation.
If I understand right, the goal is to show that for any $A$ and $B$, there is a $C$ such that $C \times A \to B$ (i.e., $(C \cap A) \subseteq B$) if and only if there is a $B^A$ such that $C \subseteq B^A$ and $(B^A \cap A) \subseteq B$. But then this doesn't seem quite right.
I'm sorry for such a beginner question. I'd really appreciate any help pointing me in the right direction or maybe clarifying what/how one would go about showing that this category has exponentiation - or what does "$B^A$" mean in terms of these subsets? Does some sequence of operations construct it?