Adjunctions via Reflections and the Axiom of Choice

Question

I have met two ways of defining adjunctions: via the triangle identities, and via reflections.

Proposition 3.1.2 Let $F:\mathsf A \rightarrow \mathsf B$ be a functor and $B$ an object of $\mathsf B$. When the reflection of $B$ along $F$ exists, it is unique up to isomorphism.

Proposition 3.1.3 Consider a functor $F:\mathsf A \rightarrow \mathsf B$ and assume that, for every object $B\in \mathsf B$, "the" reflection of $B$ along $F$ exists and such a reflection $(RB,\eta _B)$ has been chosen. In that case, there exists a unique functor $R:\mathsf B\rightarrow \mathsf A$ satisfying the two properties

1. $\forall B\in \mathsf B \; R(B)=R_B$,

2. $(\eta _B :B\rightarrow FRB)_{B\in \mathsf B}$ is a natural transformation.

Definition 3.1.4 A functor $R:\mathsf B \rightarrow \mathsf A$ is left adjoint to the functor $F:\mathsf A \rightarrow \mathsf B$ when there exists a natural transformation $\eta :1_{\mathsf B} \dot{\rightarrow}F\circ R$ such that for every $B\in \mathsf B$, $(RB,\eta _B)$ is a reflection of $B$ along $F$.

I don't know much about different axioms of choice, so I'd like a clarification of the following excerpt from the first volume of the Handbook of Categorical Algebra:

It is an immediate consequence of $3.1.2$ that in the situation of $3.1.4$, both $R$ and $\eta$ are defined uniquely up to isomorphism. On the other hand if you allow in your underlying set theory a sufficiently powerful axiom of choice, you can even conclude that a functor $F:\mathsf A \rightarrow \mathsf B$ has a left adjoint if and only if each object of $\mathsf B$ admits a reflection along $F$ (for each $B\in \mathsf B$ choose such a reflection and apply $3.1.3$.

Questions:

1. What exactly does it mean that $R$ and $\eta$ are defined up to isomorphism?
2. What kind of axiom of choice would suffice for the equivalence mention in the excerpt?
3. Is the axiom above usually assumed by category theorists, or should I be careful and not use the equivalence?

Answer 1

2. The axiom of choice for sets as big as the "sets" of objects of your categories will suffice. If you work in a framework where the objects of a category may form a proper class, then I think you need what is called "global choice". If you work with universes, then the objects of a category will form a (possibly large) set, and the ordinary axiom of choice will suffice.

3. This much choice is typically assumed by the working mathematician. In any event, in most practical cases you can actually write down an explicit reflection functor, so no choice is needed.

(1.) Try writing down what you think this might mean. You will probably be correct. Otherwise, you can probably ask a more specific question about it :).