$\require{AMScd}$I've been told that the Gray tensor product of two 2-categories can be obtained with a Kan extension (forgive me for my carelessness of foundational issues): the product $\boldsymbol\otimes \colon 2\textbf{-Cat} \times 2\textbf{-Cat} \to 2\textbf{-Cat}$ can be obtained with a Kan extension of $\Phi$ along $J$:
$$
\begin{CD}
\{\text{some 2-category}\} @>\Phi>> 2\textbf{-Cat}\\
@VJVV \\
2\textbf{-Cat} \times 2\textbf{-Cat}
\end{CD}
$$
Is this true? If yes, can you provide a reference with an *explicit* proof, or provide a proof yourself?


Thanks!



The Gray tensor product as a Kan extension
This is the subject of an [unpublished manuscript of Ross Street](http://maths.mq.edu.au/~street/GrayTensor.pdf). Street applies the technique for left Kan extending a monoidal structure along a dense functor from Brian Day's PhD thesis to construct the lax Gray tensor product of 2-categories. He defines a monoidal structure on the full subcategory $\mathbf{\text{Cu}}$ of $\mathbf{2\text{-Cat}}$ whose objects are the "cubes", and shows that the inclusion $K \colon \mathbf{\text{Cu}} \longrightarrow \mathbf{2\text{-Cat}}$ is dense. The lax Gray tensor product is then the left Kan extension of $K \circ \otimes$ along $K \times K$ as in the following diagram (which commutes up to a natural isomorphism).


$$
\require{AMScd}
\begin{CD}
\mathbf{\text{Cu}} \times \mathbf{\text{Cu}} @>K\times K>> \mathbf{2\text{-Cat}}\times\mathbf{2\text{-Cat}}\\
@V\otimes VV @VV\otimes V\\
\mathbf{\text{Cu}} @>>K> \mathbf{2\text{-Cat}}
\end{CD}$$ 


Note that the inclusion $K$ becomes a strong monoidal functor.


~~No doubt by suitably altering the monoidal category $\mathbf{\text{Cu}}$, one could also witness in this manner the pseudo Gray tensor product as a left Kan extension.~~\*


\*(Edit: This last comment is almost certainly false, since the altered category I imagine I had in mind is not dense in $\mathbf{2\text{-Cat}}$.)


