Proposition 6.3.6.8. Let $f: X \rightarrow S$ be a morphism of simplicial sets which admits a section $u: S \hookrightarrow X$. Suppose that $u \circ f$ and $\operatorname{id}_{X}$ belong to the same connected component of the simplicial set $\operatorname{Fun}_{/S}( X, X)$. Then $f$ is universally localizing.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $W$ be the collection of all edges $w$ of $X$ such that $f(w)$ is degenerate in $S$. Since our hypothesis is stable under the formation of pullbacks, it will suffice to show that $f$ exhibits $S$ as a localization of $X$ with respect to $W$. Fix an $\infty $-category $\operatorname{\mathcal{C}}$; we wish to show that composition with $f$ induces a bijection
\[ \alpha : \pi _0( \operatorname{Fun}( S, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( X[W^{-1}], \operatorname{\mathcal{C}})^{\simeq } ). \]
The injectivity of $\alpha $ follows immediately from the existence of the section $u$. To prove surjectivity, it will suffice to show that for every object $g \in \operatorname{Fun}( X[W^{-1}], \operatorname{\mathcal{C}})$ is isomorphic to $g \circ u \circ f$. Since $u \circ f$ and $\operatorname{id}_{X}$ belong to the same connected component of $\operatorname{Fun}_{/S}(X,X)$, it suffices to observe that postcomposition with $g$ carries every edge of $\operatorname{Fun}_{/S}(X,X)$ to an isomorphism in the $\infty $-category $\operatorname{Fun}( X, \operatorname{\mathcal{C}})$. $\square$