# Factorization of a map from a contractible Kan complex through a Kan complex

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.