Kerodon

Remark 9.2.1.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a morphism $w: X \rightarrow Y$. Let $\pi : \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, so that $w$ can be identified with an object $\widetilde{Y} \in \operatorname{\mathcal{C}}_{X/}$ satisfying $\pi ( \widetilde{Y} ) = Y$. Then an object $C \in \operatorname{\mathcal{C}}$ is $w$-local if and only if, for every object $\widetilde{C} \in \operatorname{\mathcal{C}}_{X/}$ satisfying $\pi ( \widetilde{C} ) = C$, the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{X/} }( \widetilde{Y}, \widetilde{C} )$ is contractible. This follows from the criterion of Remark 3.4.0.6, since $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{X/} }( \widetilde{Y}, \widetilde{C} )$ can be identified with the homotopy fiber of the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, C ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, C )$ over the vertex corresponding to $\widetilde{C}$ (see Corollary 4.6.9.18).