Coinduction sends injective abelian groups to injective modules

The formula $(r\phi )(s)=\phi (sr)$ satisfies $r_1(r_2\phi )=(r_1{r}_2)\phi$, the distributive laws, and $1\phi =\phi$, so it defines a left $R$-module structure.

For every left $R$-module $M$, evaluation at $1$ defines ${\mathrm{\Theta }}_M:{\mathrm{Hom}}_R(M,{\mathrm{Hom}}_{\mathbb{Z}}(R,D))\to {\mathrm{Hom}}_{\mathbb{Z}}(M,D),{\mathrm{\Theta }}_M(F)(m)=F(m)(1)\mathrm{.}$

The inverse sends $g:M\to D$ to $\widehat{g}(m)(r)=g(rm)$. Indeed, $\widehat{g}$ is $R$-linear because $\widehat{g}(sm)(r)=g(rsm)=(s\widehat{g}(m))(r)$, and evaluation at $1$ and the unit law show that the two constructions are inverse.

Let $u:A\to B$ be a monomorphism and $F:A\to {\mathrm{Hom}}_{\mathbb{Z}}(R,D)$. Under ${\mathrm{\Theta }}_A$, it corresponds to a group homomorphism $g:A\to D$, which extends along the underlying subgroup inclusion to $\tilde{g}:B\to D$ because $D$ is injective.

The inverse construction of step 3.1 turns $\tilde{g}$ into an $R$-linear extension $\widehat{\tilde{g}}:B\to {\mathrm{Hom}}_{\mathbb{Z}}(R,D)$ of $F$. Hence the coinduced module is injective by [F1].