The cohomological universal-coefficient extension map

AC makes $B_{n-1}C$ and $Z_{n-1}C$ free by Boundaries and cycles in a free complex over a PID are free, hence projective by Free modules are projective, with the exact choice boundary. The displayed short exact sequence is therefore a length-one projective resolution of $H_{n-1}C$. By Ext can be computed from any projective resolution of the first variable, using AC to meet its DC premise and the supplied balanced-Ext data, its degree-one cohomology identifies ${\mathrm{Ext}}_R^1(H_{n-1}C,G)$ with ${\mathrm{Hom}}_R(B_{n-1}C,G)$ modulo restrictions of maps $Z_{n-1}C\to G$.

For $\psi :B_{n-1}C\to G$, define the degree-$n$ cochain $\psi d_n:C_n\to G$. It is a cocycle because $d_n{d}_{n+1}=0$. If $\psi$ changes by $g{\mathrm{\mid }}_{B_{n-1}C}$ for $g:Z_{n-1}C\to G$, projectivity of $B_{n-2}C$ splits $C_{n-1}\twoheadrightarrow B_{n-2}C$, giving a projection ${\pi }_{n-1}:C_{n-1}\to Z_{n-1}C$. Then $(g{\mathrm{\mid }}_B)d_n=(g{\pi }_{n-1})d_n$ is a coboundary, since ${\pi }_{n-1}{d}_n=d_n$. Thus $[\psi ]\mapsto [\psi d_n]$ descends from the quotient in step 1.1 to the claimed cohomology group.

For a chain map $u:C\to D$, the pullback of $\psi d_n^D$ equals $(\psi u_{n-1}{\mathrm{\mid }}_{B_{n-1}C})d_n^C$ because $u$ commutes with differentials. This is the map induced on the two projective presentations. Postcomposition with a coefficient map $G\to G^{\mathrm{\prime }}$ also commutes exactly with the construction. Hence the resulting map is natural in the complex and in $G$; the projection used in step 2.1 establishes descent but is absent from the map itself.