Let $\mathsf{SymMonCat}$ denote the category with objects symmetric monoidal categories and morphisms lax symmetric monoidal functors. Can $\mathsf{SymMonCat}$ be endowed with a "tensor" product that makes it a (symmetric) monoidal category?
I've been thinking that maybe the product between categories could be a candidate, but I can't find anywhere something to back up this claim (probably because it isn't quite true). Any comments, references and answers are very welcome.