**Proof.**
We will use that $\text{Mod}_{(A, \text{d})}$ is an abelian category with arbitrary direct sums and products, and that these give rise to direct sums and products in $K(\text{Mod}_{(A, \text{d})})$. See Lemmas 22.4.2 and 22.5.4.

Let $M_ j$ be a family of differential graded $A$-modules. Consider the graded direct sum $M = \bigoplus M_ j$ which is a differential graded $A$-module with the obvious. For a differential graded $A$-module $N$ choose a quasi-isomorphism $N \to I$ where $I$ is a differential graded $A$-module with property (I). See Lemma 22.21.4. Using Lemma 22.22.3 we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(M, N) & = \mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(M, I) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(M_ j, I) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(M_ j, N) \end{align*}

whence the existence of direct sums in $D(A, \text{d})$ as given in part (2) of the lemma.

Let $M_ j$ be a family of differential graded $A$-modules. Consider the product $M = \prod M_ j$ of differential graded $A$-modules. For a differential graded $A$-module $N$ choose a quasi-isomorphism $P \to N$ where $P$ is a differential graded $A$-module with property (P). See Lemma 22.20.4. Using Lemma 22.22.3 we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(N, M) & = \mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M_ j) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(N, M_ j) \end{align*}

whence the existence of direct sums in $D(A, \text{d})$ as given in part (3) of the lemma.
$\square$

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