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 ?

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) By mimicking the construction of the canonical model structure on the category of small categories, 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 ?

EDIT: The discrete model structure on the category of $\Delta$-generated spaces is only accessible, not combinatorial because the underlying category is not locally finitely presentable.

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 and 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 ?

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 ?