# Interpretation of a Category theory question

A problem I'm attempting says

Let $p:A \to B$ be a map of sets and $p^*: \mathcal{P}B \to \mathcal{P}A$ be the induced map of power sets sending $X \subseteq B$ to $p^*(X) = \{a \in A: p(a) \in X\}$. Exhibit left and right adjoints to $p^*$

but I can't quite work out what it's saying the functor is: namely, are we

$(i)$ defining $p^*$ to be the functor's action on any given map $p$ between any two sets, i.e. $(-)^*$ our functor takes a function $p$ to a function $p^*$ (and $p$ is an object, not a morphism - perhaps morphisms would be commutative squares of functions), or

$(ii)$ defining $p^*$ as a functor which acts as the power set operator on sets $(A \mapsto \mathcal{P}(A))$ for any $A$, and acts on any function between sets by $p \mapsto p^*$ (so I suppose this would be a functor from Set$^\text{op}$ to Set), or

$(iii)$ fixing sets $A$ and $B$ in advance, and just defining $p^*$ on all functions $A \mapsto B$, rather than all functions between any two sets (I am not sure that this actually works, but perhaps you can formalise a single set as a single-object category with functions to itself as morphisms, a little like you can with a single group), or

$(iv)$ none of the above?

So in essence I suppose I am asking how we actually consider $p^*$ as a functor. Are our sets fixed in advance? I don't need you to actually do the question for me, I'm sure I can do that myself once I understand what's being asked, but I would appreciate your thoughts on why the question actually means what you say it does, e.g. why the other options wouldn't work (or wouldn't be sensible).

I suspect it's meant to be $(ii)$ since $p^*(X) = p^{-1}(X)$ so this would probably behave nicely with regards to functoriality, but some confirmation would be good, since I often come across these slightly ambiguously worded category theory problems so being able to quickly interpret what the question actually means would be very useful.

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2022-07-25 20:47:21
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