Group as a Category with One Object [duplicate]

Maybe the situation becomes clearer when you consider the following statement:

A group "is the same thing" as a category with one object in which every morphism is an isomorphism.

In fact, if $G$ is a group, the corresponding category, say $C_G$, has one object $\bullet$ and the morphisms from $\bullet$ to itself are given by $G$, where the composition of two morphisms $\bullet \xrightarrow{g} \bullet$ and $\bullet \xrightarrow{h} \bullet$ is $\bullet \xrightarrow{gh} \bullet$.

Conversely, if $C$ is a category with one object $\bullet$ in which every morphism $f$ (necessarily from $\bullet$ to $\bullet$) is an isomorphism, then the set of morphisms from $\bullet$ to itself forms a group $G_C := Mor(\bullet, \bullet)$. The product of two group elements is given by the composition of morphisms. The unit element of the group is given by the identity morphism on $\bullet$ and the inverse of an element of a group is given by the inverse of the morphism (since they are all isomorphisms, this is always well-defined).

It is also worth noticing that a group is a special case of a groupoid (a category in which every morphism is an isomorphism). Namely, a group is a groupoid with one object.

Similarly, a monoid can be defined as a category with one object. It is a group if and only if every morphism is an isomorphism.