Trivial Group is Terminal Object of Category of Groups

Theorem

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

Let $\set e$ be the trivial group.

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

Let $\struct {G, \circ}$ be any group.

$!: G \to \set e$

defined by:

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

By definition, any group homomorphism is also a mapping.

Hence, there is at most one morphism $\struct {G, \circ} \to \set e$ 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 $\set e$):

$\ds ! (g) * ! (h)$

$=$

$\ds e * e$

Definition of $!$

$\ds $

$=$

$\ds e$

Definition of $*$

$\ds $

$=$

$\ds ! \paren {g \circ h}$

Definition of $!$

That is, $!$ is a group homomorphism.

Thus for all groups $\struct {G, \circ}$, there is a unique group homomorphism $!: G \to \set e$.

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

$\blacksquare$