Categories enriched over symmetric closed multicategories

We construct a machine which takes as input a locally small symmetric closed complete multicategory $\mathsf{V}$. And its output is again a locally small symmetric closed complete multicategory $\mathsf{V}\text{-}\mathcal{C}at$, the multicategory of small $\mathsf{V}$-categories and multi-entry $\mathsf{V}$-functors. An example of such $\mathsf{V}$ is provided by short spaces (vector spaces with a system of seminorms) and short maps. When the ground multicategory $\mathsf{V}$ is $\mathsf{S}\mathsf{e}\mathsf{t}$ we obtain strict 2-categories and their surroundings by iterating twice the construction of categories.