Suppose we are given a contractible Kan complex $S$ and a map of simplicial set $f : S \rightarrow T$. Under what conditions can we say that $f$ factors through the largest Kan complex $Z$ contained in $T$?

I am asking this question because it pops up in Proposition 2.2.5.7 of *Higher Topos Theory*. In the proof we are given two maps $f,g : K \rightarrow \mathcal{C}$ where $K$ is an arbritary simplicial set and $\mathcal{C}$ an $\infty$-category. We also have a a contractible Kan complex $S$ and a map $h : S \rightarrow \mathcal{C}^K$ such that $h(x) = f, h(y) = g$ where $x,y$ are two distinct vertices of $S$. Lurie claims that the map $h$ then factors through the largest Kan complex $Z$ of $\mathcal{C}^K$ and I do not see why.