There's a known adjoint pair $Cat \rightarrow MonCat \dashv MonCat \rightarrow Cat$. See here.
The question is: is there a similar construction $Multicat \rightarrow MonCat \dashv MonCat \rightarrow Multicat$ ?
My idea for the forgetful functor $U:MonCat \rightarrow Multicat$ was to send a monoidal category to a new category with the same objects and with homsets that remain the same if the codomain "has no tensors", and are mapped to the emptyset otherwise ---i.e: if the codomain "has tensors".
By "has no tensors" I mean that it's not of the form $a\otimes b$ nor $1$: it's "a single object". I'm not sure one can say this and thus why I'm asking. Maybe in some subclass of monoidal categories?
This would intuitively give a "stronger" free monoidal category in the sense that we could have morphisms between lists of different sizes. We would do this by having lists of multimorphisms, one for each object in the list of codomains $a_1\otimes a_2 \otimes\dots\otimes a_n$. (And this is what the free functor would do.)
This construction is motivated by the notion of category of contexts.
Any idea? Thanks in advance!!