It is well known that any category allow for the construction of a functor from the simplex 1-category $\Delta_0$. All of the axioms of a category translate simply in the functoriality requirements.
This leads to an ordinary functor between 1-categories ($[~\_~,~=~]_0$ denoting the closed structure of $Cat_0$, the 1-category of categories and functors)
$N_0 : Cat_0 \to [\Delta_0^{op},Set]_0 $
$N_0(X)= \Delta_0^{op} \hookrightarrow Cat_0 \xrightarrow{[~\_~,~X]_0} Cat_0 \xrightarrow{U} Set$
However, it is also possible, and seems more natural in a way (as it captures more - for instance the fundamental adjunction $d_0 \dashv i \dashv d_1 $), to define a 2-functor between 2-categories
$N : Cat \to [\Delta^{op},Cat] $
$N(X)= \Delta^{op} \hookrightarrow Cat \xrightarrow{[~\_~,~X]} Cat$
I am not too familiar with the simplicial approach. Are there references which try to shy away from $N_0$ and use $N$? Is there some well known gain from doing so, like important construct which can't be done using $N_0$ which are possible using $N$ ?