Consider the discrete combinatorial model structure on the category of $\Delta$-generated spaces: all maps are cofibrations and fibrations and the weak equivalences are the homeomorphisms. Applying Proposition A.3.2.4 of (Higher Topos Theory, Lurie), one obtains a model structure on the category of topologically enriched small categories and enriched functors such that the weak equivalences $F:C\to D$ are the local homeomorphisms such that the underlying functor $F_0:C_0\to D_0$ is essentially surjective.

Like in the non-enriched case, is this model structure unique (I mean for this class of weak equivalences) ? Is it "canonical" in this sense ?