Lemma 24.22.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The differential graded category $\textit{Mod}^{dg}(\mathcal{A}, \text{d})$ satisfies axiom (C) formulated in Differential Graded Algebra, Situation 22.27.2.
Proof. Let $f : \mathcal{K} \to \mathcal{L}$ be a homomorphism of differential graded $\mathcal{A}$-modules. By the above we have an admissible short exact sequence
\[ 0 \to \mathcal{L} \to C(f) \to \mathcal{K}[1] \to 0 \]
To finish the proof we have to show that the boundary map
\[ \delta : \mathcal{K}[1] \to \mathcal{L}[1] \]
associated to this (see discussion above) is equal to $f[1]$. For the section $s : \mathcal{K}[1] \to C(f)$ we use in degree $n$ the embeddding $\mathcal{K}^{n + 1} \to C(f)^ n$. Then in degree $n$ the map $\pi $ is given by the projections $C(f)^ n \to \mathcal{L}^ n$. Then finally we have to compute
\[ \delta = \pi \circ \text{d}_{C(f)} \circ s \]
(see discussion above). In matrix notation this is equal to
\[ \left( \begin{matrix} 1
& 0
\end{matrix} \right) \left( \begin{matrix} \text{d}_\mathcal {L}
& f
\\ 0
& -\text{d}_\mathcal {K}
\end{matrix} \right) \left( \begin{matrix} 0
\\ 1
\end{matrix} \right) = f \]
as desired. $\square$
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