The Stacks project

Exercise 111.30.13. Let $\mathcal{A}$ be an abelian category. Consider a solid diagram

of complexes of $\text{Fil}^ f(\mathcal{A})$. Assume $K^\bullet$, $L^\bullet$ and $I^\bullet$ bounded below and each $I^ n$ a filtered injective object. Also assume $\alpha$ a filtered quasi-isomorphism. Any two morphisms $\beta _1, \beta _2$ making the diagram commute up to homotopy are homotopic.