Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ satisfies something". In my case, I have something close to being the identity, yet I don't know the best approach to show it is an equivalence if I don't want to mess up too much with combinatorial stuff.


If we were in the Quillen model structure, it would be almost straightforward: up to substituting $f$ with a Kan fibration, we would have a fiber sequence $|\textrm{hofib}(f) |\to |\mathcal{C}| \to |\mathcal{D}|$ and the criterion would be "if $\textrm{hofib}(f)$ is weakly contractible, then $f$ is a weak equivalence". Does this generalize to the Joyal model structure? I have the impression I am missing some model-categorical proof since the statement is general.


PS Since I believe my $f$ is a Joyal fibration, we can ignore the eventual difference between homotopy fibers in the two contexts (I am starting to doubt the two model categories both underlie the $\infty$-category of simplicial sets).



A fiber-like method to show equivalence of infinity categories
An obvious necessary condition for $f$ to be a categorical equivalence is that $f$ is weakly equivalent to a (co)cartesian fibration of quasicategories, i.e., $f$ is an analogue of a [Street fibration](https://ncatlab.org/nlab/show/Street+fibration) of quasicategories.


Since $f$ is already a Joyal fibration, a plausible course of action is to check right away whether $f$ is a (co)cartesian fibration, i.e., has (co)cartesian lifts for morphisms.


If the above necessary condition holds, then $f$ is a categorical equivalence if and only if its fibers (which are automatically homotopy fibers because $f$ is a Joyal fibration) over every vertex of $\cal D$ are contractible Kan complexes, e.g., by Lurie's straightening-unstraightening theorem (Higher Topos Theory, Theorem 3.2.0.1, combined with Proposition 2.4.2.4).


