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. 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.

Let $\mathbf{C}$ be a quasicategory. Using a version of the Yoneda lemma for quasicategories, Joyal constructs, in Section 15.3 of his Notes on Quasi-Categories, a simplicial category $\overline{\mathbf{C}}$ such that $\mathrm{N}_\Delta(\overline{\mathbf{C}})$ is equivalent to $\mathbf{C}$.

The result is actually valid for any simplicial set.

Lemma. If

is a Quillen equivalence, then the composition $$A\xrightarrow{\eta_A}G(F(A))\xrightarrow{G\left(P_{F(A)}\right)}G(P(F(A)))$$ is a weak equivalence, where $P(A)$ denotes the fibrant replacement of an object $A\in\mathscr{A}$.

Proof. See Lemma 3.5.2 of these notes for a proof.

Now, HTT 2.2.5.1 states that $(\mathfrak{C},\mathrm{N}_\Delta)$ gives a Quillen equivalence:

The above lemma then implies that given a cofibrant object in $\mathit{sSets}_\mathrm{Joyal}$ (that is, any simplicial set) $S_\bullet$, we have a weak equivalence $$S_\bullet\rightarrow\mathrm{N}_\Delta\left(P(\mathfrak{C}[S_\bullet])\right).$$

In particular, if $\mathscr{C}$ is a quasicategory, then $Q(\mathfrak{C}[\mathscr{C}]) $ is a simplicial category whose homotopy coherent nerve is equivalent to $\mathscr{C}$.