Fix a complete, cocomplete, symmetric monoidal closed category $\mathcal{V}$. I will also assume that there is a forgetful functor $\mathcal{V}\to \mathbf{Set}$ with a left adjoint. By standard results, this adjunction lifts to a 2-adjunction between the 2-category of categories and the 2-category of $\mathcal{V}$-categories; IOW, we can speak of the free $\mathcal{V}$-category on a category.

Now, let $\mathcal{A}$ be a (small) symmetric monoidal closed $\mathcal{V}$-category. Then the $\mathcal{V}$-presheaf category $\mathbf{PShv}(\mathcal{A})$ is symmetric monoidal closed when endowed with the Day convolution structure and the Yoneda embedding is symmetric monoidal closed. Furthermore, any symmetric monoidal $\mathcal{V}$-functor $F:\mathcal{A}\to \mathcal{B}$ with $\mathcal{B}$ cocomplete, lifts uniquely (up to unique isomorphism) to a cocontinuous symmetric monoidal functor $\mathbf{PShv}(\mathcal{A})\to \mathcal{B}$ by taking the left Kan extension $L_{Y}(F)$ of $F$ along Yoneda $Y$.

I can prove all this to myself without great effort. What has left me stumped is the following question:

Q: assume $\mathcal{B}$ and $F$ closed, is $L_{Y}(F)$ closed?

Since Yoneda is closed the converse is trivially true. If the answer is negative in general, is there any criteria to have a positive answer? If it helps the answer, my use case is when $\mathcal{A}$ is the free $\mathcal{V}$-category on a Heyting algebra (e.g. the open sets of a topological space) with monoidal structure the intersection. Since in a lattice there is at most one arrow between two objects, Day convolution degenerates into the pointwise tensor product.

Thanks in advance, regards, G. Rodrigues