Question: In what sense is "the kernel of a group homomorphism a universal"?
Do we here mean that a group homomorphism is a universal arrow in some comma category? If so, what comma category?
Attempt: If we let $K$ be a normal subgroup of $G$ and $i: K \rightarrow G$ being the inclusion homomorphism, then for any other normal subgroup $T$ of $K$ with $k : T \rightarrow K$ being another inclusion homomorphism and likewise with $j : T \rightarrow G$, then we have that $j = i \circ k$.
Now let $C$ be the category of the subgroups of $G$ with inclusion homomorphisms as morphisms. Let $S: C \rightarrow C$ be the identity functor on $C$. Then do we not here have that $(K, i)$ is a universal arrow in $(C \downarrow G)$? And if so, is this the sense in which "the kernel of a group homomorphism is a universal"?
EDIT: Answers in another thread on this site have confused me in the sense that they don't discuss explicitly about universal arrows and comma categories.