When are enriched categories equivalent? $F : \mathbf{MonCat} \to \mathbf{2Cat}$ is the 2-functor for change of enrichment. What is the maximal subcategory of $\mathbf{MonCat}$ whose arrows $b : V \to W$ each induce an equivalence of categories $F(b) : V\mathbf{Cat} \cong W\mathbf{Cat}$?
My current guess is that we can take some restricted portion of the poset of embeddings of categories in $\mathbf{MonCat}$, perhaps using some sort of adjointness requirement. I convinced myself that this works with a diagram chase, but I think I'm wrong.
This question was split from a more general question on MSE about implications of identifying such subcategories.
 A: Here's a partial confirmation of Simon Henry's hunch:
Observation: Let $F: V \to W$ be a strong monoidal functor. Suppose that $V$ and $W$ have initial objects $\emptyset$ preserved by $\otimes$ in each variable separately, and preserved by $F$. Suppose that $F$ induces an equivalence $VCat \to WCat$. Then $F$ is an equivalence.
Proof: The assumptions allow us to form, for each $v \in V$, the category $\Sigma(v)$, which has two objects $0,1$, with $Hom(0,1) = v$, $Hom(0,0) = Hom(1,1) = I$, $Hom(1,0) = \emptyset$, and to observe that $\Sigma_V : V \to VCat$, $v \mapsto \Sigma v$ is a fully faithful functor. The same also holds in $WCat$. Moreover, if $F$ induces an equivalence it must carry $V$-functors which induce bijections on isomorphism classes of objects to $W$-functors with the same property. From this it soon follows that the restriction of $F$ to $V$-categories of the form $\Sigma(v)$ is inverse to the restriction of $F^{-1}$ to the $\Sigma(w)$'s, and so because $\Sigma_V$ and $\Sigma_W$ are fully faithful, it follows that $F$ is an equivalence.

Possibly the general case can be reduced to this one via Day convolution or something?

I think it's a bit more interesting to look a the the 2-category $V-Mod$ rather than $VCat$. I believe that Ross Street may have written about the analogous question here, but I'm not sure of the reference.
