Q:

How does one define constant functor of simplicial sets?

In HA, p142, Lem 1.4.2.10 Lurie makes the following construction, 

Let $C. D$ be $\infty$-categories which admits final objects. Let $D$ have final object $*'$.  Let $X:C \rightarrow D$ be the constant functor taking value $*'$. 

How is the constant functor constructed? 

A:

It's just the composite of the unique simplicial map $C\to \Delta^0$ with the map $\Delta^0\to D$ representing $*'$. Note that maps $\Delta^0\to D$ are naturally bijective with elements of $D(0)$ by Yoneda. 

