I am not aware of Kock's works.

Nevertheless Kelley provides the definition of its tensor product in the next page: it defines its tensor product $\mathcal A \otimes_{\mathcal F} \mathcal B$ as the $\mathcal F$-theory generated by the sketch $((\mathcal A \otimes \mathcal B)^{op},\Phi)$ where $\Phi$ is made of the $\mathcal A \otimes \mathcal B$ cylinder of the form $\lambda \otimes B$ and $A \otimes \mu$ defined in the previous paragraph ($\lambda$ and $\mu$ range over the family of colimit cylinders of $\mathcal A$ and $\mathcal B$ respectively).

I hope this helps.

I am not aware of Kock's works.

Nevertheless Kelly provides the definition of its tensor product in the next page: it defines its tensor product $\mathcal A \otimes_{\mathcal F} \mathcal B$ as the $\mathcal F$-theory generated by the sketch $((\mathcal A \otimes \mathcal B)^{op},\Phi)$ where $\Phi$ is made of the $\mathcal A \otimes \mathcal B$ cylinder of the form $\lambda \otimes B$ and $A \otimes \mu$ defined in the previous paragraph ($\lambda$ and $\mu$ range over the family of colimit cylinders of $\mathcal A$ and $\mathcal B$ respectively).

Edit: I see your problem was not with the tensor in $\mathcal F-\mathbf{Cat}$ the tensor $\mathcal A \otimes \mathcal B$. This is the tensor product of $\mathcal A$ and $\mathcal B$ as $\mathcal V$-categories, the definition can be found in section 1.4 page 12.

I think it is important to stress the fact $\mathcal A \otimes \mathcal B$ is not a $\mathcal F$-complete category.

I hope this helps.