Let $(\mathcal{M},\otimes)$ be a symmetric monoidal model category; I'll assume for simplicity that every object is fibrant. Suppose that the unit $I$ is NOT cofibrant. I'm interested in whether the derived tensor product with the unit is oplax/strong monoidal.

On the one hand, since $-\otimes^L I$ is the left derived functor of $-\otimes Q$ where $Q$ is a cofibrant replacement of $I$, an oplax structure on the derived functor could come from a monoidal Quillen adjunction with left adjoint $-\otimes Q$. This requires to find a "natural" diagonal map $Q\to Q\otimes Q$, and that the map $Q\otimes Q \to I$ be a weak equivalence.

On the other hand, the unit axiom seems to indicate that $-\otimes^L I$ is strong monoidal anyway, since $I\otimes^L I\simeq I$; but this isomorphism can be presented by two maps: $Q\otimes Q \to Q\otimes I$ and $Q\otimes Q \to I\otimes Q$. Does this really induce a strong monoidal structure, and are the two structures "the same" ?