The nerve functor $N:Cat\to SSet$ from the category of small categories to simplicial sets can be obtained as follows: The left Kan extension of the functor $F$ which sends $[n]$ to the category $\bullet\to\ldots\to\bullet$ along the Yoneda embedding $\Delta\to SSet$ gives an adjunction
$$
h:SSet \leftrightarrows Cat:N
$$
with $N_n(\mathcal{C})=Hom_{Cat}(F(n),\mathcal{C})$.
There exists also a simplicial nerve functor $\mathfrak{N}$ invented by Cordier, I think. Its construction can be found in Lurie's HTT. It is a functor from the category of small simplicial categories (small categories enriched over simplicial sets) $Cat_\Delta$ to $SSet$. It seems to me that $\mathfrak{N}$ can be constructed using a left Kan extension as above too, but of another functor $F$, Lurie calls ${\mathfrak{C}}[-]$ which I can not typeset correctly. Is this true?
Can $\mathfrak{N}$ be constructed as follows also? Maybe it's the same thing:
There exists an enriched version of left Kan extensions. The categories $SSet$ and $Cat_\Delta$ are canonically enriched over $SSet$. The Yoneda embedding is a simplicial functor if $\Delta$ is considered as a discrete simplicial category. Maybe ${\mathfrak{C}}[-]$ is a simplicial functor too and perhaps it can be defined more naturally in this way, I don't know. Then one would get an enriched adjunction
$$
sh:SSet \leftrightarrows Cat_\Delta:R
$$
and I wonder if $R$ is $\mathfrak{N}$ then.
Best Answer
About the first question: Yes, the simplicial nerve is an instance of the same general construction which gives you the usual nerve (and e.g. also the Quillen equivalences between simplicial sets and topological spaces, between models for the $\mathbb{A}^1$-homotopy category given by simplicial presheaves and sheaves respectively and much more):
A cosimplicial object in a cocomplete category E gives, via Kan extension, rise to an adjunction between $E$ and simplicial sets, where the left adjoint goes from simplicial sets to $E$ and comes from the universal property of the Yoneda embedding (namely that functors from $C$ to $E$, $E$ cocomplete, are the same as colimit perserving functors from $Set^{C^{op}}$ to $E$). This is (part of) Proposition 3.1.5 in Hovey's book on model categories.
Lurie uses the same pattern; it is enough to give a cosimplicial object in simplicial categories, which is done in his definition 1.1.5.1.