Let $i: X \to \mathrm{N}\mathcal C$ be a monomorphism in the category of simplicial sets, with $\mathcal C$ a category and $\mathrm{N}\mathcal C$ its nerve. I am looking for sufficient conditions (and not too difficult to check) under which this monomorphism is a Joyal equivalence.
A sufficient condition I know is to apply $\mathfrak C$, the adjoint of the homotopy coherent nerve and check that $\mathfrak C(i)$ is a Dwyer-Kan equivalence.
Since $\mathcal C$ is a category, $\mathfrak C \mathrm N\mathcal C$ is DK-equivalent to the discrete simplicial category $C$, thus one should find conditions under which $\mathfrak CX$ is also DK-equivalent to the discrete simplicial category $\mathcal C$.
For the $\pi_0$ condition, one can use the fact that $(\pi_0)_* \mathfrak C \cong h$, where $h$ is the adjoint to the classical nerve, and relatively well understood.
Is there any known way of computing the fundamental group and/or the homology of the hom spaces of $\mathfrak C X$?
