Example 10.3.5.15. Let $\operatorname{\mathcal{C}}$ be a regular $\infty $-category. Then, for every object $X \in \operatorname{\mathcal{C}}$, the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ is also regular (Proposition 10.3.5.9). Moreover, for every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, Proposition 7.6.2.24 guarantees that there exists a functor
\[ f^{\ast }: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/X} \quad \quad Z \mapsto X \times _{Y} Z \]
given by pullback along $f$. The functor $f^{\ast }$ is regular: it preserves finite limits since it right adjoint to the postcomposition functor $(f \circ \bullet ): \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/Y}$, and preserves quotient morphisms by virtue of Proposition 10.3.5.3.