The cohomology universal-coefficient sequence splits nonnaturally

The exact sequence $0\to Z_{n}C\to C_n\stackrel{d_n}{\to }{B}_{n-1}C\to 0$ comes from The cycle-boundary short exact sequences for a free complex over a PID. Under AC, Boundaries and cycles in a free complex over a PID are free makes $B_{n-1}C$ free and Free modules are projective, with the exact choice boundary makes it projective. Lift its identity through $d_n$ to choose a section $s$. Then $1-s{d}_n$ factors through the inclusion $Z_{n}C\to C_n$ as a projection $\pi :C_n\to Z_{n}C$ restricting to the identity on cycles.

Write $q:Z_{n}C\to H_{n}C$ for the quotient. For $f:H_{n}C\to G$, define $\sigma (f)=[fq\pi ]$. The map $fq\pi :C_n\to G$ is a cocycle: $d_{n+1}$ lands in $B_{n}C\subseteq Z_{n}C$, where $\pi$ is the identity and $q$ is zero. This assignment is $R$-linear in $f$, so it defines a homomorphism into cohomology.

For a cycle $z$, $(fq\pi )(z)=f(qz)$, so ${\mathrm{ev}}_n\sigma (f)=f$. Thus $\sigma$ is a section of the surjection in the exact UCT sequence. Its construction uses the chosen projection; it supplies existence without asserting naturality in $C$.