In the book Algebra: Chapter 0, the author asks the following problem.
Let $G$ be a group, $H\trianglelefteq G$ and $N$ is subgroup of $G$ with $H\subseteq N$.
(Exer. 8.11, p. 107) Prove by hand (i.e. without invoking universal properties) that $N$ is normal in $G$ if and only if $N/H$ is normal in $G/H$.
I know how to do this; but my question is other side of the problem.
Invoking universal properties, how can we prove this?