I am trying to understand double categories and their relatives a little better. I do not understand much, so I apologize in advance if my question is too naive for this website.
A functor $F:X \rightarrow Y$ of ordinary $2$-categories is an equivalence if and only if:
(1) It is essentially surjective.
(2) It is fully faithful, in the sense that it induces equivalences of Hom categories.
Are there analogues of conditions (1) and (2) that detect equivalences of double categories?
To ask my question in a different way, section A.3.2 of Lurie's Higher Topos Theory constructs a model structure on $S$-enriched categories where $S$ is a monoidal model category satisfying some nice properties (an alternative set of properties were recently discovered by Berger and Moerdijk). In Lurie's model structure, an $S$-enriched functor $F:X \rightarrow Y$ is an equivalence if and only if
(1) $Map_X(x,x') \rightarrow Map_Y(F(x),F(x'))$ is always a weak equivalence in $S$
(2) $F$ induces an essentially surjective map of homotopy categories $Ho(X) \rightarrow Ho(Y)$. Here, arrows from $x$ to $x'$ in $Ho(X)$ are homotopy classes of morphisms from the unit of $S$ to $Map_X(x,x')$.
Suppose I have a nice model category $S$, for some definition of nice I am willing to determine. Can I expect there to be a model structure on category objects in $S$, analogous to the above model structure on $S$-enriched categories? What would be the analogues of conditions $(1)$ and $(2)$?