On the abstract of [a paper by Emily Riehl and Dominic Verity](https://arxiv.org/pdf/1808.09834.pdf), it is stated that



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> Every quasicategory arises as a the homotopy coherent nerve of a simplicial category up to equivalence.
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> 
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Where can one find a proof of this statement?




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On motivation for the question, HTT 4.2.4.1 would then imply that any colimit in a quasicategory could be computed as a homotopy colimit of a diagram $F\colon\mathbf{J}\rightarrow\mathbf{C}$, where $\mathbf{J}$ and $\mathbf{C}$ are simplicial categories.



Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category
Sorry, we should have made this more clear. One proof appears as Theorem 7.2.2 in our previous paper in this series, [The comprehension construction](https://www.math.jhu.edu/~eriehl/comprehension.pdf). As suggested by others, it follows from a suitably defined Yoneda embedding (which is hard to construct in the ∞-categorical setting).


Explicitly we show that a quasi-category B is equivalent to the homotopy coherent nerve of the full simplicial subcategory of the slice category qCat/B spanned by the right fibrations B/b -> B for each vertex b in B. Because the only objects we consider here are right fibrations the hom-spaces between two such in qCat/B are automatically Kan complexes.


The hard part of this is defining the map of quasi-categories from B to the homotopy coherent nerve, which by adjunction we construct as a simplicially enriched functor indexed by the "homotopy coherent realization" of B.


