Kerodon

Proof. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, and let us identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. By virtue of Corollary 9.2.7.13, it will suffice to show that if $\widetilde{f}$ is a morphism in $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f} ) = f$, then $\widetilde{X}$ is $\widetilde{f}$-local. Choose a colimit diagram $Q: K^{\triangleright } \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ which satisfies $Q|_{K} = Q_0$ and carries the cone point of $K^{\triangleright }$ to $f$. Since the inclusion of the cone point into $K^{\triangleright }$ is right anodyne (Example 4.3.7.11) and the projection map $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_{/Y} ) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ is a right fibration (Corollary 4.2.5.2) we can lift $Q$ to a diagram $\widetilde{Q}: K^{\triangleright } \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ carrying the cone point to $\widetilde{f}$. Using Corollary 7.1.4.21 and Proposition 7.1.7.3, we see that $\widetilde{Q}$ is a colimit diagram which is preserved by the evaluation functors $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_{/Y} ) \rightarrow \operatorname{\mathcal{C}}_{/Y}$. For each vertex $v \in K$, let $\widetilde{f}_{v} = \widetilde{Q}(v)$ is a morphism of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f}_{v} ) = f_{v}$. Our assumption that $f_{v}$ is left orthogonal to $g$ guarantees that $\widetilde{X}$ is a $\widetilde{f}_{v}$-local object of the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$ (Corollary 9.2.7.13). Applying Remark 9.2.1.9, we deduce that $\widetilde{X}$ is also $\widetilde{f}$-local. $\square$