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.