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.

First note that $$\mathcal C^K$$ is again an $$\infty$$-category. Then this follows from the fact that passing from an $$\infty$$-category to the largest Kan complex contained in it is a functor right adjoint to the inclusion of Kan complexes into all $$\infty$$-categories. This is proved in Prop 1.2.5.3 of HTT.
That is, adjointness tells us more generally that if $$S$$ is a Kan complex and $$X$$ is an $$\infty$$-category, then any map $$S \to X$$ factors through the largest Kan complex contained in $$X$$. (The ordinary categorical counterpart of this statement is that if $$S$$ is a groupoid and $$X$$ is a category, then any functor $$S \to X$$ factors through the maximal subgroupoid of $$X$$, i.e. the groupoid of isomorphisms in $$X$$. In the same way, the 1.2.5.3 says that the largest Kan complex contained in an $$\infty$$-category $$X$$ is given by throwing out all 1-cells which are not equivalences.)
• It’s maybe worth saying explicitly: “the largest Kan complex contained in A” isn’t generally well-defined, for an arbitrary simplicial set A. There’s a right adjoint giving “the terminal (algebraic) Kan complex over A”, but in general this isn’t a subobject of A. So trying to justify why “the largest Kan complex contained in $C^K$” is well-defined can lead one naturally to this answer. – Peter LeFanu Lumsdaine Jan 5 at 1:20