I have been reading the paper by Toën "The homotopy theory of dg categories and derived Morita theory" where in chapter 4 it is stated that the tensor product of two cofibrant dg categories $C$ and $D$ (defined by $obj(C\otimes D)=objC\times obj D$ and with space of morphisms obtained by tensoring the two chain complexes in $C$ and $D$) is not in general cofibrant, and this is a problem in both the "standard" and the Morita model structures on the category $\mathbf{dgCat}_k$ (here $k$ is a commutative ring with 1 and with arbitrary characteristic).

It is also known (proposition 2.3 (2) in the same article) that a cofibrant replacement functor can be chosen in such a way that it is the identity on objects, therefore the problem seems to lie exclusively in what happens to morphisms. The same proposition contains the result that a cofibrant dg category has morphisms spaces which are cofibrant complexes wrt the projective model structure on $\mathbf{Ch}(k)$.

A cofibrant chain complex $X_\bullet$ has $X_n$ projective $\forall n$ and the converse is true only if $X_\bullet$ is bounded below. Nevertheless, the tensor product (and the direct sum) of two projective modules is again projective so I think I am missing some essential point: given $X_\bullet$ and $Y_\bullet$ cofibrant chain complexes, $(X\otimes Y)_\bullet$ will be made of projective modules, so I think I need to find an unbounded cofibrant complex which tensored with some other cofibrant complex loses cofibrancy.

This result was cited in several other places, but I was unable to find even a discussion longer than the statement. I would be glad if someone could give me such an example, or point out some reference where I can find it.

Thank you in advance