# Problem with Notation in Category Theory

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?

• 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.) – Derek Elkins Feb 12 at 5:07
• $1$ 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. – Lord Shark the Unknown Feb 12 at 5:08
• @DerekElkins that makes a lot of sense. – Zack Ni Feb 12 at 5:12
• @LordSharktheUnknown Thank you! – Zack Ni Feb 12 at 5:14

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, "$$!_U$$").