Is there a monoidal structure on the category $Spc^{\Delta}$ of cosimplicial spaces such that in the adjunction $$\Delta^{\bullet}\otimes-\colon Spc\leftrightarrows Spc^{\Delta}\colon Tot(-)$$ the left adjoint is (strong) symmetric monoidal? If this is true, can it be generalized by replacing $Spc$ by any symmetric monoidal category?