Yes, I see there are other Q&A's on this, for instance here: Projective objects in the category of chain complexes

I am wondering why a level-wise projective chain complex $P$ which is split exact can be written as a direct **sum** of the $P(n)$ from Ralph's first answer in the link above, rather than as a direct product. If $P$ is unbounded in both directions, for instance, then couldn't $P$ contain an element $(\dots, p_{-1},p_0, p_1, \dots )$ that is a priori not contained in $\bigoplus_{n\in \mathbb{Z}} P(n)$?

In the reference [DS95], Dwyer & Spalinski impose the condition that the chain complex be non-negatively graded [i.e., study projective objects in $Ch_{\geq 0}(\mathcal{A})$. I do not want to impose this condition: I am interested in projective objects in $Ch(\mathcal{A})$, that are unbounded. For that matter, even in the bounded below case $Ch_{\geq 0}(\mathcal{A})$, I don't understand why $P$ cannot have an element $p= (\dots, 0, \dots, 0, p_0, p_1, p_2, \dots )$ which could be zero in negative degrees and non-zero in all non-negative degrees, which is not an element of $\bigoplus_{n\in\mathbb{Z}}P(n)$.

[DS95] = Dwyer & Spalinski, Homotopy theories and model categories.