# 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.