But stephen lack's "A 2-categories companion", page 4, has the following interesting passage: (emphasis mine)
Suppose V and W are monoidal categories and F : V → W is a left adjoint
which does preserve the monoidal structure up to coherent isomorphism.
There is no reason why the right adjoint U should do so, but there will be
induced comparison maps UA ⊗ UB → U(A ⊗ B) making U a monoidal
functor. (Think of the tensor product as a type of colimit, so the left adjoint
preserves it, but the right adjoint doesn’t necessarily.) In fact the monoidal
functor U : Ab → Set arises in this way