Question on Adjoints

Question

I am working on the following question:

(a) Let $\mathscr{C}$ and $\mathscr{D}$ be complete categories and let $J$ be a small category. Suppose that: $F:\mathscr{C}\to\mathscr{D}$ and $G:\mathscr{D}\to\mathscr{C}$ satisfies that $F\dashv G$; $F_*:\mathscr{C}^J\to\mathscr{D}^J$ and $G_*:\mathscr{D}^J\to\mathscr{C}^J$ form the induced adjunction $F_*\dashv G_*$; $\triangle:\mathscr{C}\to\mathscr{C}^J$ and $\lim\limits_J: \mathscr{C}^J\to\mathscr{C}$ is the limit adjunction, and, by abuse of notation, $\triangle:\mathscr{D}\to\mathscr{D}^J$ and $\lim\limits_J: \mathscr{D}^J\to\mathscr{D}$ is also the limit adjunction. Prove that $\lim\limits_J\circ G_*$ and $G\circ\lim\limits_J:\mathscr{D}^J\to\mathscr{C}$ are naturally isomorphic.

(b) Suppose that $f:X\to Y$ is a set map, and consider the induced image functor $f:\mathcal{P}(X)\to\mathcal{P}(Y)$ between the power set of $X$ and the power set of $Y$. Does the latter $f$ have a left adjoint? Prove or disprove.

I think (a) can be proven based on the uniqueness of adjoints (up to natural isomorphisms), but I am not sure how that is achieved (I did a proof and the notations got messed up easily, so I'm not sure if I'm on the right track)? I don't really have a clue about (b) though, I am aware that it has a right adjoint, which is the map to preimages... appreciate your help!

Answer 1

(a) is the well-known theorem that right adjoints preserve limits.

For (b), if $f_* : P(X) \to P(Y)$ is a right adjoint, then it preserves limits. But limits here just intersections, so we get $f_*(\bigcap_i A_i) = \bigcap_i f_*(A_i)$. For the empty index set, this means $f_*(X)=Y$, i.e. $f$ is surjective. We can also take binary intersections of singletons: $f_*(\{x\} \cap \{x'\}) = f_*(\{x\}) \cap f_*(\{x'\})$, which means $f(x)=f(x') \implies x=x'$, so $f$ has to be injective. So if $f_*$ has a right adjoint, then $f$ is bijective. Of course, the converse is true as well.

Answer 2

You're right about (a), that is, you solve it by the uniqueness of adjoint. In particular, if you have two adjoint pairs $F\dashv G:\mathscr C\rightarrow\mathscr D$ and $X\dashv Y:\mathscr C\rightarrow\mathscr E$, then $FX$ is left adjoint to $YG$, since $$\mathbf{Hom}(FX-,-)\simeq\mathbf{Hom}(X-,G-)\simeq\mathbf{Hom}(-,YG-)$$and the equivalences are all natural.

In particular, $\text{lim}_J\circ G_*$ is right adjoint to $F_*\circ\triangle$, while $G\circ \lim_J$ is the right adjoint to $\triangle\circ F$. Simply apply the definitions to prove that $F_*\circ\triangle=\triangle\circ F$, then your done: $G\circ \lim_J$ and $\text{lim}_J\circ G_*$ are isomorphic, being right adjoint to the same functor.