Kleisli biadjunction is symmetric lax monoidal
Prove that, for a linear exponential pseudocomonad on a symmetric Gray monoid K with Kleisli bicategory K^{oc}, the Kleisli biadjunction K ⊣ J between K^{oc} and K becomes a symmetric lax monoidal adjunction when K is equipped with the tensor product ⊗ and K^{oc} with the cartesian product ×. Equivalently, show that the left biadjoint K is symmetric strong monoidal by establishing that the Seely equivalences are pseudonatural with respect to Kleisli maps.
References
We conjecture that the Kleisli biadjunction in~equ:cokleisliadjunction becomes a symmetric lax monoidal adjunction when $K$ is considered as a symmetric Gray monoid with respect to $\otimes$, and ${K}$ is considered as a symmetric monoidal bicategory with respect to $\times$, as in
\begin{equation*}
\begin{tikzcd}column sep = large
\ar[r, shift left =1, bend left =10, "K"]
\ar[r, description, phantom, "\scriptstyle \bot"] &
(K, \otimes, I) \mathrlap{.}
\ar[l, shift left = 1, bend left =10, "J"]
\end{tikzcd}
\end{equation*}
As yet we have not fully verified the pseudonaturality conditions.