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.
Proof. For every $i \in I$ choose a (termwise) injective map of complexes $K_ i^\bullet \to L_ i^\bullet $ which is homotopic to zero with $L_ i^\bullet $ quasi-isomorphic to zero. For example, take $L_ i^\bullet $ to be the cone on the identity of $K_ i^\bullet $. We define $\mathbf{M}^\bullet (M^\bullet )$ by the following pushout diagram
Then $M^\bullet \to \mathbf{M}^\bullet (M^\bullet )$ is a functor. The right vertical arrow defines the functorial injective map $j_{M^\bullet }$. The cokernel of $j_{M^\bullet }$ is isomorphic to the direct sum of the cokernels of the maps $K_ i^\bullet \to L_ i^\bullet $ hence acyclic. Thus $j_{M^\bullet }$ is a quasi-isomorphism. Part (2) holds by construction. $\square$
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