FALSE: enough projectives imply a canonical resolution for every object

One projective resolution of $\mathbb{Z}\mathrm{/}2$ is $0\to \mathbb{Z}\stackrel{2}{\to }\mathbb{Z}\to \mathbb{Z}\mathrm{/}2\to 0.$ Adding the contractible projective complex $0\to \mathbb{Z}\stackrel{1}{\to }\mathbb{Z}\to 0$ in degrees $1$ and $0$ gives a different projective resolution $0\to \mathbb{Z}\oplus \mathbb{Z}\stackrel{\mathrm{diag}(2,1)}{\to }\mathbb{Z}\oplus \mathbb{Z}\to \mathbb{Z}\mathrm{/}2\to 0,$ where the augmentation is reduction modulo $2$ on the first summand. Both displayed augmented complexes are exact, but they are not the same resolution.

Thus even in a category with enough projectives the property alone does not uniquely determine a resolution of a fixed object. Moreover, [L2] shows that arbitrary objectwise choices still do not uniquely determine the comparison maps of a resolution functor. The special construction in [L3] uses the extra underlying-set structure of a module category, while [L1] records that a general supplied datum is additional structure. Therefore the displayed claim is false.