I'm reading Fokkinga's Gentle Introduction to Category Theory, of which page 12 asks to give categorically expressed properties (i.e., using the language of basic CT and existensial qualifiers) in $\textbf{Set}$ for, among others:
- $P(A) \quad\equiv\quad A = \{17\}$
- $P(A) \quad\equiv\quad A$ is a singleton set.
Am I correct in thinking that the former is impossible -- namely because you can't "look inside" an object using only category theory tools? Whereas for the second, $A$ is a singleton set iff for all $X$ in $\textbf{Set}$ there is only one arrow $X \to A$?
(The rest of the mini-exercise is rather confusing, as well. I'd be forever grateful if someone could supply an "answer sheet" for when I get stuck!)