Lemma 24.25.12. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $R$ be a set and for each $r \in R$ let an injective map $\mathcal{M}_ r \to \mathcal{M}'_ r$ of acyclic differential graded $\mathcal{A}$-modules be given. There exists a functor $M : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{A}, \text{d})$ and a natural transformation $j : \text{id} \to M$ such that

$j_\mathcal {M} : \mathcal{M} \to M(\mathcal{M})$ is injective and a quasi-isomorphism,

for every solid diagram

\[ \xymatrix{ \mathcal{M}_ r \ar[r] \ar[d] & \mathcal{M} \ar[d]^{j_\mathcal {M}} \\ \mathcal{M}'_ r \ar@{..>}[r] & M(\mathcal{M}) } \]

a dotted arrow exists in $\textit{Mod}(\mathcal{A}, \text{d})$ making the diagram commute.

**Proof.**
We define $M(\mathcal{M})$ as the pushout in the following diagram

\[ \xymatrix{ \bigoplus _{(r, \varphi )} \mathcal{M}_ r \ar[r] \ar[d] & \mathcal{M} \ar[d] \\ \bigoplus _{(r, \varphi )} \mathcal{M}'_ r \ar[r] & M(\mathcal{M}) } \]

where the direct sum is over all pairs $(r, \varphi )$ with $r \in R$ and $\varphi \in \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A}, \text{d})}(\mathcal{M}_ r, \mathcal{M})$. Since the pushout of an injective map is injective, we see that $\mathcal{M} \to M(\mathcal{M})$ is injective. Since the cokernel of the left vertical arrow is acyclic, we see that the (isomorphic) cokernel of $\mathcal{M} \to M(\mathcal{M})$ is acyclic, hence $\mathcal{M} \to M(\mathcal{M})$ is a quasi-isomorphism. Property (2) holds by construction. We omit the verification that this procedure can be turned into a functor.
$\square$

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