Universal Property of Quotient of Topological Group

Theorem
Let $G$ and $H$ be topological groups.

Let $N$ be a normal subgroup of $G$.

Let $\pi : G \to G/N$ be the quotient mapping.

Let $f : G \to H$ be a continuous group homomorphism whose kernel contains $N$.

Then there exists a unique continuous group homomorphism $\overline f : G/N \to H$ such that $f = \overline f \circ \pi$.

Proof
Because $N\subset \operatorname{ker} f$, $f$ is constant on the cosets of $N$.

Thus $f$ is invariant under congruence modulo $N$.

By Universal Property of Quotient Set, there exists a unique mapping $\overline f : G/N \to H$ such that $f = \overline f \circ \pi$.

It suffices to verify that it is a continuous group homomorphism.

By Universal Property of Quotient Space, there exists a continuous mapping $\overline g : G/N \to H$ such that $f = \overline g \circ \pi$.

By uniqueness of $f$, $g=f$.

By Universal Property of Quotient Group, there exists a group homomorphism $\overline h : G/N \to H$ such that $f = \overline h \circ \pi$.

By uniqueness of $f$, $h=f$.