I am confused with the exclamation mark used in Steve Awodey's book Category Theory. The following diagram is from page $98$:

$$ \require{AMScd} \begin{CD} U @>\boxed{\,!\;}\ \color{red}{???}>> 1 \\ @VVV @VV{\top}V \\ A @>>{\chi_{U}}> 2. \end{CD} $$

The book says that

Therefore, we can rephrase the correspondence between subsets $U \subset A$ and their characteristic function $\chi_U:A \to 2$ in terms of pullbacks: (the picture above)

The thing I am not understanding is because that he used the notion of $!$ in the book first time (or maybe I did not read carefully enough). I do not know what he meant by exclamation?

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    $\begingroup$ It's just an arrow that's named $!$. In particular, it is the unique arrow into $1$ given by its universal property. (Often $!$ will be subscripted to be clearer, e.g. $!_U$, but the subscript is often omitted.) $\endgroup$ – Derek Elkins Feb 12 at 5:07
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    $\begingroup$ $1$ is a terminal object in the category of sets. By $!$ (or $!_U$) he means the unique map from the object $U$ to the terminal object. $\endgroup$ – Lord Shark the Unknown Feb 12 at 5:08
  • $\begingroup$ @DerekElkins that makes a lot of sense. $\endgroup$ – Zack Ni Feb 12 at 5:12
  • $\begingroup$ @LordSharktheUnknown Thank you! $\endgroup$ – Zack Ni Feb 12 at 5:14

In the interest of having an answer to this question (with a tip o' the hat to both Derek Elkins and Lord Shark the Unknown for their comments):

In the diagram $$ \require{AMScd} \begin{CD} U @>{!}>> 1 \\ @VVV @VV{\top}V \\ A @>>{\chi_{U}}> 2, \end{CD} $$ the top arrows is labeled "$!$", which (naïvely) indicates that the arrow is simply named "$!$". That being said, the name is not without motivation: exclamation points are often used in mathematics to denote uniqueness. For example, you might often see the notation "$\exists !x$", which is read "there exists a unique $x$".

In the context of category theory, 1 is a terminal object in the category of sets. That is, if $U$ is any set, then there exists a unique morphism from $U$ to 1. In notation, we might write $$U \in \operatorname{Ob}(\mathsf{Set}) \implies \exists ! f:U\to 1. $$ As the morphism $f$ is unique, we may as well label it "$!$" (or, if there is a risk of ambiguity, "$!_U$").


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