Proposition 5.4.3.9. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ be the projection map. Let $Y$ be an object of the $(\infty ,2)$-category $\operatorname{\mathcal{C}}_{/f}$, and let $\overline{u}: \overline{X} \rightarrow q(Y)$ be a morphism in the $(\infty ,2)$-category $\operatorname{\mathcal{C}}$. Then $\overline{u}$ can be lifted to a morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}_{/f}$ with the following property:
- $(\ast )$
For every vertex $z \in K$, the image of $u$ in $\operatorname{\mathcal{C}}_{/f(z)}$ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.