Lemma 13.19.6. Let $\mathcal{A}$ be an abelian category. Consider a solid diagram

where $P^\bullet $ is bounded above and consists of projective objects, and $\alpha $ is a quasi-isomorphism.

There exists a map of complexes $\beta $ making the diagram commute up to homotopy.

If $\alpha $ is surjective in every degree then we can find a $\beta $ which makes the diagram commute.

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