UCT collapse gives a natural splitting

Take $C_1=\mathbb{Z}a\oplus \mathbb{Z}b$, $C_0=\mathbb{Z}c$, $da=2c,db=0$, and $M=\mathbb{Z}\mathrm{/}2$. Then $H_{0}C=\mathbb{Z}\mathrm{/}2$, $H_{1}C=\mathbb{Z}b$, and the Hom cochain differential is $m\mapsto (2m,0)=0$. Hence $H^1\mathrm{Hom}(C,M)=M^2$. F2 makes its quotient onto $\mathrm{Hom}(H_{1}C,M)=M$ the map $(x,y)\mapsto y$. Its kernel is $M\oplus 0$, the Ext term of F1. The map $y\mapsto (0,y)$ is a section, so existence is not the issue.

The chain automorphism $a\mapsto a+b$, $b\mapsto b$, $c\mapsto c$ fixes $H_{0}C,H_{1}C$ and therefore both end terms. On Hom cohomology it acts by $(x,y)\mapsto (x+y,y)$. Every section must lift $1\in M$ to $(x,1)$, which this automorphism moves. Naturality would require that lift to be fixed, a contradiction. Thus even an existing splitting of a collapsed two-column UCT need not be natural. The general UCT invocation inherits AC for PID free-submodule arguments; this particular finite cochain and shear calculation uses no choice.