Day convolution of two V-functors $F, G$ is defined as
$$(F \ast G)(X) = \int^{A,B} \mathbf{C}(A \otimes B, X) \otimes F(A) \otimes G(B).$$
This can be also seen as $F \ast G = \mathsf{Lan}_{\otimes}(F \otimes G)$.
What would be a reasonable dual of this concept?
The second definition hints at something like $\mathsf{Ran}_{(\to)} (F \to G)$ for $(\to)$ the exponential. However, I do not see how could this be unital, for instance. What are the options if we want something like a dual of Day convolution and which properties do they satisfy?