Lemma 13.19.7. Let $\mathcal{A}$ be an abelian category. Consider a solid diagram

$\xymatrix{ K^\bullet & L^\bullet \ar[l]^\alpha \\ P^\bullet \ar[u] \ar@{-->}[ru]_{\beta _ i} }$

where $P^\bullet$ is bounded above and consists of projective objects, and $\alpha$ is a quasi-isomorphism. Any two morphisms $\beta _1, \beta _2$ making the diagram commute up to homotopy are homotopic.

Proof. Dual to Lemma 13.18.7. $\square$

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