why does unique identity make groups pointed sets? The content is from Aluffi Algebra Chapter 0 (p43) .

I can't understand what it means.

this makes groups pointed sets

I think I can have pointed sets just with group definition.
Or I misunderstand the notion of pointed set(group) .
Could you give me an example of pointed set of group?

Example 3.7:
he mention "
coslice category: a category C from a ﬁxed object A to all objects in C. "
The following is Example 3.8.


 A: The fact that a group has a unique identity means that there can only be one element $e$ in that group such that $e \cdot x = x = x \cdot e$ for all $x$ in the group. This is what Proposition 1.6 in your screenshot says.
A pointed set is a pair $(X, x)$ where $X$ is a set and $x \in X$ is an element. A morphism of pointed sets $f: (X, x) \to (Y, y)$ is a function $f: X \to Y$ such that $f(x) = y$.
So every group $G$ forms a pointed set in a natural way, namely as $(G, e_G)$ where $e_G$ is the identity element of $G$. Any group homomorphism $f: G \to H$ then forms a morphism of pointed sets as $f: (G, e_G) \to (H, e_H)$ since group homomorphisms preserve identity elements: $f(e_G) = e_H$.
In particular this means that we have a functor $\mathbf{Grp} \to \mathbf{Set^*}$ from the category of groups to the category of pointed sets.

Relating this to the second screenshot: we may view a pointed set $(X, x)$ also as a function $p: \{*\} \to X$ with $p(*) = x$. Writing out definitions you will see that morphisms of pointed sets are then 'the same thing' as morphisms in that coslice category.
