FALSE: objectwise projective-resolution choices uniquely determine a resolution functor

The proof of [L1] is objectwise: it chooses terms and differentials but supplies no unique lift of a morphism between resolved objects.

The exact row $0\to \mathbb{Z}\stackrel{2}{\to }\mathbb{Z}\to \mathbb{Z}\mathrm{/}2\mathbb{Z}\to 0$ is a projective resolution by [L2]. On two copies of it, multiplication by $1$ in both degrees and multiplication by $3$ in both degrees are distinct chain maps lifting the identity of $\mathbb{Z}\mathrm{/}2\mathbb{Z}$: both commute with multiplication by $2$, and $3\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}2)$. Thus the chosen objectwise resolution does not uniquely determine the map assigned to the identity morphism.

Therefore objectwise resolution choices do not uniquely determine comparison maps or a resolution functor; additional coherent choices or a separate functorial construction are required.