Proposition 4.5.1.8. The inclusion functor $\mathrm{h} \mathit{\operatorname{Kan}} \hookrightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ of Remark 4.5.1.4 admits a left adjoint.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We wish to show that there exists a Kan complex $X$ and a morphism $u: \operatorname{\mathcal{C}}\rightarrow X$ with the following property: for every Kan complex $Y$, precomposition with $u$ induces a bijection
\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}} }( X, Y) = \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}}}( X, Y) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{QCat}}}( \operatorname{\mathcal{C}}, Y ). \]
Unwinding the definitions, we see that this is a reformulation of the requirement that $u$ is a weak homotopy equivalence of simplicial sets. The existence of $u$ now follows from Corollary 3.1.7.2. $\square$