Here a simplicial category is a simplicial object in $\textbf{Cat}$ (that is, a functor from $\Delta^{op}$ to $\textbf{Cat}$). I wonder why the nerve of a simplicial category is a simplicial set?
For a simplicial category, say, $X$, $X_n$ is a category, and when we apply the nerve to $X$, each $N(X_n)$ is a simplicial set, which is just a functor, not a set, I wonder why $N(X_*)$ is a simplicial set?
Best Answer
The nerve of a simplicial category, as you observe, is most naturally a bisimplicial set. One can then extract various simplicial sets from here, such as the diagonal. Whatever author you're reading should be doing something to explain how the nerve of a simplicial category is being viewed as a simplicial set.