Let $V$ and $W$ be symmetric monoidal categories. Let $F:V\to W$ be a lax symmetric monoidal functor with multiplication $\nabla:FA\otimes FB \to F(A\otimes B)$. Consider the following statements:

1) There are lax symmetric monoidal functors $F(-)^{\otimes n}:V\to W$ and $F((-)^{\otimes n}):V\to W$ for each $n\geq 2$,

2) There is a lax monoidal transformation $F(-)^{\otimes n} \Rightarrow F((-)^{\otimes n})$.

They seems plausible to me. For example, if $n=2$ these things are a long but doable diagram chase. As $n$ gets bigger, it gets more and more unwieldy to actually verify it by hand.

It seems to me this is a sort of "coherence theorem". For example, consider the second statement for $n=3$. There are two reasonable natural transformations to define by suitably tensoring $\nabla$ with the identity, yet they are equal by associativity of $F$. As $n$ grows, the amount of "obvious but provably equal" options for defining the natural transformation grows.

Are these statements true, and how might one go about actually proving them?

Note: there is a hidden (iterated) diagonal which might be a red herring. We could be considering the functors in 1) to be $V^{\times n}\to W$.