Torsion-free abelian groups form a reflective full subcategory of abelian groups

If nonzero integers $m,n$ kill $x,y\in G$, then $mn$ kills $x+y$, and $m$ kills $-x$; hence $\mathrm{Tor}(G)$ is a subgroup. If $n(g+\mathrm{Tor}(G))=0$ in the quotient for nonzero $n$, then $ng$ is torsion, so some nonzero $m$ has $mng=0$; since $mn\mathrm{\ne }0$ in $\mathbb{Z}$, $g$ is torsion. Thus the quotient is torsion-free.

If $f:G\to H$ and $H$ is torsion-free, every torsion element $x$ has nonzero $n$ with $nf(x)=f(nx)=0$, so [L1] gives $f(x)=0$. Hence $\mathrm{Tor}(G)\subseteq \ker f$, and [L2] gives a unique $\bar{f}:G\mathrm{/}\mathrm{Tor}(G)\to H$ through which $f$ factors.

Step 2.1 supplies, for every abelian group $G$, the quotient map $G\to G\mathrm{/}\mathrm{Tor}(G)$ together with the unique factorisation of any map into a torsion-free group; that is exactly a specified universal arrow from $G$ to the inclusion. By the equivalence in [L4] these supplied arrows assemble into a reflector $R$ with $R⊣I$, which by [L3] says the full torsion-free subcategory is reflective, with unit the quotient map.