In a series of papers ([\[1\]][1], [\[2\]][2] and [\[3\]][3]), Dwyer and Kan introduced the *Hammock localization* [[2]] as an effective technique to compute the simplicial localization of a model category [[1]]. This is meant to encode the higher homotopy implicit in a model category; indeed, if one takes $\pi_0$ hom-wise the result is the homotopy category of the original model category. 

They say their main application is to recontruct the function complexes between fibrant cofibrant objects in a simplicial model category. And they managed. Cool! On the other hand, a feature-oriented description of such a localization is given in [[3]]. Unfortunately, at that time (I guess!) the Lurie technology was not available, so a universal-property localization was not so easy to state in such a context. But now we do have it, so I wonder:

> Given a model category $C$ with weak equivalences $\mathcal{W}$, consider its Hammock Localization $L^H(C, \mathcal{W})$. Take its derived homotopy coherent nerve $\textbf{R}N_{coh}(L^H(C,\mathcal{W}))$ and you will get a quasicategory. On the other hand, you can take the ordinary nerve of $C$, and then invert $\mathcal{W}$ in the sense of Lurie ([6], 1.3.4.1), obtaining $N(C)[\mathcal{W}^{-1}]$. Do these two constructions agree?

As a recall, the derived homotopy coherent nerve is obtained by first taking a fibrant replacement (aka locally kan simplicial category) and then applying the homotopy coherent nerve (which will automatically be fibrant in $\textrm{sSets}$, i.e. a quasicategory). 

**The furthest I could get.** There is a [nice paper by Hinich][4] in which some of these issues are discussed. In particular, if $C$ is a fibrant simplicial (say model) category with weak equivalences $\mathcal{W}$, there is a weak equivalence of marked simplicial sets (proposition 1.2.1):
$$ (**) \ \ \ (N_{coh}C, \mathcal{W}) \to \textbf{R}N(L^H(C,\mathcal{W}))^{\natural} $$
Notice also that the right-hand side is fibrant in the category of marked simplicial sets (over the point): indeed, from HTT 3.1.4.1 one can easily deduce that fibrant here means beaing a quasicategory with equivalences as marked edges. Also, in remark 1.3.4.2. from HA, we see that $N_{coh}C[\mathcal{W}^{-1}]$ can be identified with a fibrant replacement of $(N_{coh}C,\mathcal{W})$ in marked simplicial sets. Since equation (**) gives an explicit fibrant replacement, we get the thesis. 

Unluckily, proposition 1.2.1 from Hinich is available only in the case in which $C$ is a (fibrant) simplicial category, though it's not clear where the simplicial structure is used. In fact, hammock localization is designed (also) to provide a simplicial structure on an arbitrary model category when such an extra structure does not come for free. 
Thanks for your help! It's not really my field and I find it hard sometimes to orient in the choose-your-model yoga.

**Bibliography**

 - [[1]] - Simplicial localization of categories, *Dwyer and Kan*
 - [[2]] - Calculating simplicial localization, *Dwyer and Kan*
 - [[3]] - Function Complexes in Homotopical Algebra, *Dwyer and Kan*
 - [[4]] - DK localization revisited, *Vladimir Hinich*
 - [5] - Higher Topos Theory, *Jacob Lurie*
 - [6] - Higher Algebra, *Jacob Lurie*

  [1]: https://pdf.sciencedirectassets.com/271593/1-s2.0-S0022404900X02095/1-s2.0-0022404980900493/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEHgaCXVzLWVhc3QtMSJHMEUCIQCJfF2x4pwX8axP9HZDJjSiD%2FpgUfsTnw83IjdyRFx2vAIgW8UIr3Q0nFlbTwizj7ijnKkYMQftJn09pB%2Bq5Dd8aSUq%2BgMIQBAEGgwwNTkwMDM1NDY4NjUiDAjs5QiRFLh7m5NWLSrXA31wzOonCx2A8Pg1kCQZ43ElH4WGghSD8W4hnfr9mfBECseJdxJQtftjTMnnTdyDZca%2B2vQCbWpJlbrnnCKzhaxVfZkPJ2L%2BK%2F3BE6Js3t4brglz1XuemOMRVx0COoh8bII4GQX%2FhUoH7Satp5Qp1zIPrF30AgU9V9x28kV6q5NdsYfOjWHomq9sg8kTOrhtrjZ61nKThNHndbFsK6fg2ksPlPcf87QNrwI8o8YAd8XUUELT4lehlgVTBo7sMmcvxIY0MYM9Xt8v8bC%2Bk60vOhwGepyqpJQoSMC9UrmFKuOSANslTZ7U3eqFuwV4niHfWl5KprOrLJS5gGyWq5dz94k3%2FatdJSb0Gp5gYUr3GAIpdbOhoy1lBFidKClYqFPozBGzFH0f79zwmkA4i%2Fyuc8zWsF4VVXzrCUM4WNFwwUeI73D3pq%2BUzz2eYrKV21Rvs8fhLdfZWVqzOHhm%2FFzsi7UKMttn78%2FB%2BEwSCMtiKFZTuNE0CJ4LSaeJyqM37uakUI92g%2B5aXOw5ohZwk%2B%2B09nRrJ%2FfLTHVpji8%2BU5jKqYD89dv6ZWWQbHv0DUzveMiuhLCD5wwx8SvKRuxyQ261pf0luIKI1q1lR3R%2FpCANb%2BNX2Cw2qhOACTCYxqaGBjqlAbElrV7lS0FHGYT%2FggObnUqWohJ2Vo13ZriXazlhigQKrxTe4yZvIpGAhmOYxq5xwEfPlI3fS%2Fr1J%2BY%2FOoIRXTuMXorfUGwqyqTNyYPisJtpn6tqAtv135U4jd%2BmhC%2F7TY87juXI1kB9uAFLeCX6dgixi7uDCLpjQKmOhtSQShgLmUZ76F6ZoMzPpAZFoTOXONbkaCYdw9rXFBSsImFYAFRoPtIjhw%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20210616T073959Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYQZNPMSTJ%2F20210616%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=a4fa28cfeb37c530d40189f51a775dbbec0bd9dccfaba52a27f30f19c81a3536&hash=1710a5f750d8a247350cf807bd3ff63899cd87f060cf0bff6cfe480e522ed0a9&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=0022404980900493&tid=spdf-91d79469-70c8-413e-9709-b8dd30b97332&sid=ecac2a237a47e04f422b7257b78c8473b263gxrqb&type=client
  [2]: https://www3.nd.edu/~wgd/Dvi/CalculatingSimplicialLocalizations.pdf
  [3]: https://people.math.rochester.edu/faculty/doug/otherpapers/dwyer-kan-3.pdf
  [4]: https://arxiv.org/abs/1311.4128v4