When are enriched categories equivalent?

$F : mathbf{MonCat} to mathbf{2Cat}$ is the 2-functor for change of enrichment. Is there a subcategory of $mathbf{MonCat}$ whose arrows $b : V to W$ each induce an equivalence of categories $F(b) : Vmathbf{Cat} cong Wmathbf{Cat}$?

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.