Trivial Group is Terminal Object of Category of Groups

Theorem
Let $\mathbf{Grp}$ be the category of groups.

Let $\left\{{e}\right\}$ be the trivial group.

Then $\left\{{e}\right\}$ is a terminal object of $\mathbf{Grp}$.

Proof
Let $\left({G, \circ}\right)$ be any group.

By Singleton is Terminal Object, there is precisely one mapping:


 * $!: G \to \left\{{e}\right\}$

defined by:


 * $\forall g \in G: ! (g) = e$

By definition, any group homomorphism is also a mapping.

Hence, there is at most one morphism $\left({G, \circ}\right) \to \left\{{e}\right\}$ in $\mathbf{Grp}$.

Now to verify that the mapping $!$ is a group homomorphism.

For any $g, h \in G$, we have (using $*$ for the group operation on $\left\{{e}\right\}$):

That is, $!$ is a group homomorphism.

Thus for all groups $\left({G, \circ}\right)$, there is a unique group homomorphism $!: G \to \left\{{e}\right\}$.

That is, $\left\{{e}\right\}$ is a terminal object of $\mathbf{Grp}$.