Do functions whose domains are infinite sets sequentially or simultaneously map their elements Here are two equivalent definitions of the axiom of choice

Let $x$ be a set. Suppose that if $y,w \in x$, then $y \neq \varnothing$ and $y\cap w = \varnothing$. Then there is a set $z$ such that if $y \in x$, then $y \cap z$ contains a single element.

and

Let $I$ be a nonempty, indexing set and let $\{A_i\}_{i\in I}$ be a family of nonempty sets indexed by $I$. Then there is a function $f: I \rightarrow \bigcup_{i \rightarrow I} A_i$ such that $f(i) \in A_i$ for all $i \in I$.

The text says with regards to the first definition that if we want to choose one element from each set in an infinite family of nonempty sets, we must make the choices simultaneously instead of sequentially. I would like to know where in either definition it can be understood that the elements are chosen at the same time rather than one-by-one, or is this just mere terminology as to be didactily correct yet mathematically irrelevant.
The author also mentions that in proving that if $f: A \rightarrow B$ is surjective, then $f$ has a right inverse, one element $a \in A$ is chosen simultaneously such that $b = f(a)$, which leads me to also wonder if whenever we say, "Let $a \in A$ such that $b = f(a)$", we mean that the elements in $A$ are being simultaneously mapped by $f$ to a $b$; and if that's true, then when would $f$ sequentially map the elements in $A$ to some of the elements in $B$? 
I know my issue with understanding how the elements are chosen is so trivial as to be a mere distracting quibble, but I really want some clarification. Note that this is not an axiom of choice question, since I'm really just curious as to how the elements are selected and in what ways do we distinctly want elements to be sequentially or simultaneously chosen.
 A: The difference between choosing things "simultaneously" and choosing them "sequentially" is that when you choose them sequentially, your later choices are allowed to depend on your previous ones. This is the difference between countable choice and dependent choice. 
But this has nothing to do with functions. A function's values are fixed in "Platonic reality"; they aren't "chosen" in any sense. 
A: If $A$ is a non-empty set, to choose an element of $A$ means that we have some new constant symbol $a$ and we assert that $a\in A$. This is also known as existential instantiation. We prove, or assume that $\exists x(x\in A)$, then we can "instantiate" the existential quantifier and obtain "an actual element of $A$".1
We can repeat this finitely many times, so this essentially means that we can choose from finitely many sets at once.2
Using the axiom of choice, in however formulation of it that you want, given a family of sets $\{A_i\mid i\in I\}$ there is a function, $f$ such that $f(i)\in A_i$. This means that we have a term describing an element of $A_i$. Since $f$ does it for all $i\in I$ it means that simultaneously we chose elements from each $A_i$.
If you want to really get formal, then the family of sets is itself a function $A$ with domain $I$ such that for all $i\in I$, $A(i)$ is a non-empty sets. But at this point it becomes an obstruction to the intuitive understanding of the term "choose simultaneously from all the sets in $\{A_i\mid i\in I\}$", so I'll stop.



*

*(with quotation marks, because what does it even mean an actual element of $A$?)

*(I am cheating you here, from a mathematically correct point of view; but intuitively this is how I think about that, and how I explain my students when we first talk about AC.)

A: The axiom of choice chooses somewhat magically. So I doubt either a sequential description or a simultaneous one would give a good picture of how the choosing operates. For example:
$I=\mathbb{R}$ and $\{A_i\}_{i\in I}$ where $A_i=\mathbb{R}$. The axiom of choice states that there exist a choice function $f:I\to \bigcup_{i\in I}A_i$ where $f(i)\in A_i$.  (Note that $\bigcup_{i\in I}A_i=\mathbb{R}$)
But what this specific description I gave is saying, is that there exist a function with domain $\mathbb{R}$ and range $\mathbb{R}$. So a list of choice functions could be: 


*

*$f(x)=x$

*$f(x)=\sin{x}$

*$f(x)= e^x$

*$\int e^{x^2}dx$


etc. Being that I wouldn't describe the the function $f(x)=e^x$ as a sequential choice of numbers on the real line, I don't think sequential choice making is the best description. (I would simply view  $e^x$ as more of a magical description that if you ask for the value of $x$ at $f(x)$ I can tell you, but neither you nor I have the time to go through and look at them all.)

Here is the thing with the axiom of choice. You don't know how the choices are being made. You don't know how they look like. You just assume that you can make one. That is it. So asking how a choice is being made is pointless unless you are talking about finitely, or countably many choices, which are easier to deal with.
