I have two major questions.
In the following diagrams, only some of the arrows make sense to me or are explained. Others I have never seen and are not brought up in any of the webpages or books that I have seen.
We are working within the ambient category $\mathrm{Set}$.
$\underline{1}$ denotes the singleton set.
Rather than stating which parts of the diagram commute or using a $\checkmark$ symbol, we will use coloring: Any red or blue path may commute with a black path (and will count as a red/blue path still). However, no red path may commute with a blue path.
(1)
I know that the arrow $\underline{1} \xrightarrow{\quad a \quad} A$ categorizes the notion that $a \in A$.
Completely unanswered in my texts is what the arrow $A \xrightarrow{\quad ?\quad} \underline{1}$ is intended to mean. What does this arrow mean in $\mathrm{Set}$? Why would it even need to be here?
I don't want any tired recital of "pullbacks" or "spans" or any other category theory jargon that I can already see on wikipedia, I want to know what sort of concrete mapping this is meant to be in set theory, why it needs to be here for this particular instance, and not what it is in terms of abstract nonsense.
(2)
I think that I know what a monic arrow (also called a monomorphism) is. This is an arrow $A \xrightarrow{\quad j \quad} X$ where $j(a) = j(a') \Longrightarrow a = a'$.
The internal structure of $\underline{2}$ can be given in the diagram:
Let's look at set $S = \{\alpha, \beta\}$ and subset $A = \{\alpha \}$. For this toy example, I want every membership to be explicitly categorized. Per my rules, would this be the correct diagram that gives "all" the information?
Then can we state by reading the arrows off the diagram that
$\chi_A (\alpha) = \chi_A (j (\alpha)) = \mathrm{True}$