There are a number of Grothendieck constructions: one for discrete categories, one for enriched categories (see Tamaki's paper here) and one for quasicategories (see the Unstraightening and Straightening correspondence of Lurie's Higher Topos Theory that goes between certain model categories whose underlying quasicategories are the ones of simplicial sets over a fixed simplicial set, and functors from that simplicial set to the quasicategory of quasicategories). Moreover, there is a construction that takes a simplicially enriched category whose morphism simplicial sets are Kan complexes and produces a quasicategory, namely the homotopy coherent nerve. My question is about how the coherent nerve construction interacts with the enriched Grothendieck construction and the Grothendieck construction on simplicial sets described by Lurie.
Now, suppose I have a pseudofunctor from a diagram category to simplicially enriched categories $F:I\to sCat$. The enriched Grothendieck construction gives me a functor $C\to I$, where $C$ is a simplicially enriched category. We can extend this functor along the homotopy coherent nerve $hN:sCat\to sSet$, and by Proposition 184.108.40.206 of Higher Topos Theory, we know that there is a corresponding morphism $X\to N(I)$ where $X$ is a simplicial set. How does this object relate to the coherent nerve applied to $C$, i.e. the morphism of simplicial sets $N(C)\to N(I)$, where $I$ is thought of as a simplicially enriched category by thinking of its morphism sets as 0-dimensional simplicial sets?