Universal property of the quotient group

Statement

Let $G$ and $H$ be groups, let $N$ be a normal subgroup of $G$, and let $\pi \colon G \to G/N$ be the projection $\pi(g) = gN$ to the quotient group. Let $\varphi \colon G \to H$ be a group homomorphism with $N \subseteq \ker \varphi$. Then there is exactly one group homomorphism $\bar{\varphi} \colon G/N \to H$ with $\bar{\varphi} \circ \pi = \varphi$, and it is given by $\bar{\varphi}(gN) = \varphi(g)$.

Remarks

The map $\bar{\varphi}$ is well defined because $\varphi$ is constant on each coset of $N$. The image of $\bar{\varphi}$ is the image of $\varphi$, and the kernel of $\bar{\varphi}$ is $(\ker \varphi)/N$. With $N = \ker \varphi$, this gives the first isomorphism theorem.

Explanations

L1 human

A homomorphism that sends all of $N$ to the identity cannot tell apart elements of the same coset of $N$. So it already makes sense on the cosets, that is, on the quotient group. This is the standard way to define a homomorphism out of a quotient group.