How does one define constant functor of simplicial sets?

Question

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?

Answer 1

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.