Let's look at the universal property of quotient groups:
Let $\varphi:G \to H$ be a homomorphism, $N$ a normal subgroup of $G$ and $\pi:G \to G/N$ the canonical projection. If $N \le \ker \varphi$, there is a homomorphism $\tilde \varphi: G/N \to H$ such that $$\tilde \varphi \circ \pi = \varphi.$$
The proof in my book for this universal property states that the uniqueness of $\tilde \varphi$ follows directly from the surjectivity of the canonical projection $\pi$.
However, i don't understand, how and why.
How would a comprehensive proof of the uniqueness of $\tilde \varphi$ look like, rather than stating "follows from surjectivity"? In other words:
Why would $\tilde \varphi$ not be unique if $\pi$ wasn't surjective?
Thanks for any help!