Lemma 19.12.4. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $(K_ i^\bullet )_{i \in I}$ be a set of acyclic complexes. There exists a functor $M^\bullet \mapsto \mathbf{M}^\bullet (M^\bullet )$ and a natural transformation $j_{M^\bullet } : M^\bullet \to \mathbf{M}^\bullet (M^\bullet )$ such

$j_{M^\bullet }$ is a (termwise) injective quasi-isomorphism, and

for every $i \in I$ and $w : K_ i^\bullet \to M^\bullet $ the morphism $j_{M^\bullet } \circ w$ is homotopic to zero.

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