Corollary 10.2.5.6. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces. Then $\operatorname{\mathcal{S}}$ is a regular $\infty $-category.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

**Proof.**
Corollary 7.4.5.6 guarantees that $\operatorname{\mathcal{S}}$ admits finite limits. By virtue of Corollary 10.2.5.5, it will suffice to show that every map of Kan complexes $f: X \rightarrow Y$ factors as a composition $i \circ q$, where $q: X \twoheadrightarrow Y_0$ is a universal quotient morphism in $\operatorname{\mathcal{S}}$ and $i: Y_0 \hookrightarrow Y$ is a monomorphism in $\operatorname{\mathcal{S}}$. For this, we can take $i: Y_0 \hookrightarrow Y$ to be the inclusion of the essential image of $f$ (which is a monomorphism by Example 9.2.4.10), and $q: X \rightarrow Y_0$ to be the restriction of $f$ (which is a universal quotient morphism by Proposition 10.2.4.17.
$\square$