Let $\text{DGA}^{-}$ denote the category of non-positively graded differential graded algebras with differentials of degree $+1$. It is well-known that $\text{DGA}^{-}$ has a model structure with

- Weak equivalences are quasi-isomorphisms;
- Fibrations are morphisms that are surjective on all negative degrees;
- Cofibrations are the morphisms having the left-lifting property with respect to acyclic fibrations.

It is also clear that $\text{DGA}^{-}$ has a tensor product which makes it a monoidal category.

My question is: Is the model structure compatible with the tensor product? More precisely, is $\text{DGA}^{-}$ a monoidal model category with the above model structure and tensor product?