# In what sense is the kernel of a group homomorphism a universal? [duplicate]

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.

• @StefanPerko: Answers in that thread have confused me in the sense that they aren't discussing explicitly about universal arrows and comma categories. – user1770201 Mar 11 '17 at 11:23
• The question is still very similar and I'm not sure you are really asking the question you should be asking ("How do you translate between the different ways to express universal properties?", something along those lines). – Stefan Perko Mar 11 '17 at 11:33