'Uniqueness' of adjoint functors

Question

Let $F:\mathcal{A}\to\mathcal{B}$ be a left-adjoint functor. What 'choices' need to be made in order to construct a right-adjoint $G:\mathcal{B}\to\mathcal{A}$ for $F$ and a natural isomorphism $$\text{Hom}(F(-),-)\to\text{Hom}(-,G(-)).$$ With category theory, I tend to think of something as 'unique' only when it is unique up to unique isomorphism, yet in some texts on category theory I have seen references simply to the existence of some natural transformation or isomorphism in relation to adjoints, which indicates that some choices are required i.e. the construction of the right-adjoint is not completely canonical. I am particularly interested in the case of when a symmetrical monoidal category is closed in this context (see my earlier question).

Answer 1

Going through the three standard approaches to adjunctions:

• Constructions of the adjunction (which is not the same as merely constructions of $G$) correspond to choices of objects $Gb$ for $b\in\mathcal{B}$ and terminal morphisms $\epsilon_b:FGb\to b$
• Constructions of the adjunction correspond to choices of objects $Gb$ and bijections of sets $\mathcal{B}(Fa,b)\cong\mathcal{A}(a,Gb)$ which are natural in $a$ (choices of representation of the functor $\mathcal{B}(F,b)$ as Jerry says)
• Constructions of the adjunction correspond to choices of functors $G$ and natural transformations $\eta,\epsilon$ satisfying the triangle identities

The last one is definitely the worst perspective to take here since it makes it seem like you have to do a lot of work, a lot of construction, whereas the first two - especially the first - are very minimalist by comparison.

Adjoints are unique up to unique isomorphism... in a certain sense of unique. Maybe it is better to say they are "canonical" up to "canonical" isomorphism.

Given two right adjoints $(G,\sigma:\mathcal{B}(Fa,b)\cong\mathcal{A}(a,Gb))$ and $(G',\tau:\mathcal{B}(Fa,b)\cong\mathcal{A}(a,G'b))$, there is a unique isomorphism of the pair $(G,\sigma)\cong(G',\tau)$ in the sense that there is a unique natural isomorphism $\lambda:G\cong G'$ such that $\mathcal{A}(1,\lambda_b)\sigma_{a,b}=\tau_{a,b}$; there is a unique isomorphism that also respects the adjunction.

Answer 2

Let $F:\mathsf C\to \mathsf D$ be a left adjoint functor.

Given a right adjoint $G\vdash F$, by definition $Gd$ (for all objects $d$ in $\sf D$) is a representing object of $\mathsf D(F-,d):\sf C\to Set$.

On the other hand, fixed (for all objects $d$ in $\sf D$) a representing object $r_d$ of $\mathsf D(F-,d)$, the map $d\mapsto r_d$ extends to one and only one functor right adjoint to $F$.

Hence you get to choose how the right adjoint behaves on the objects, but after that its behavior on the morphisms is fixed; moreover for any choice of two functors $G,G'\vdash F$, there is a natural isomorphism $G\cong G'$.