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.