Universal morphism in the first isomorphism theorem for groups.

Question

I almost don't know anything about categories but I was reading about universal morphisms and I wanted to see this in the context of the first isomorphism theorem for groups. What would the functor $F$ be in this case?

Answer 1

Let $K$ be a normal subgroup of $G$, and consider the category $\mathcal C$ whose objects are the homomorphisms $\varphi:G\to H$ which vanish on $K$: $\varphi(K)=\{1_H\}$, and whose morphisms are commutative triangles (this is a full subcategory of the comma category $G/\mathcal{Grp}$).

Then define $F:\mathcal C\to\mathcal{Grp}$ by sending $\varphi:G\to H$ to $H$.