It goes like this:
Let $G$ be a group, $H < G$, and $N \lhd G$ be the smallest normal subgroup containing $H$. Then any f \in \operatorname{Hom}(G,G') such that $H$ is in its kernel uniquely factors through $G/N$.
It's a simple proposition, but it's so weirdly formulated that I spent the whole day picking it apart.
Basically, it says two things:
1) $H < N \lhd G$, and $N < \operatorname{Ker}f$ (so $N \lhd \operatorname{Ker}f$), which is trivial to prove.
2) Any hom uniquely factors through a quotient by any normal group that is a subgroup of its kernel.
So why combine two natural propositions into a single one that is so weird? Is it used in some important place or does it have any significance in the group theory by itself? Or am I mistaken and the second proposition I put forward is generally false?