Corollary 9.6.6.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. Let $U: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, so that $g$ can be identified with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $U( \widetilde{X} ) = X$. The following conditions are equivalent:
- $(1)$
The morphism $f$ is left orthogonal to $g$ (in the sense of Definition 9.6.6.1).
- $(2)$
For every morphism $\widetilde{f}: \widetilde{A} \rightarrow \widetilde{B}$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $U( \widetilde{f} ) = f$, the object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ is $\widetilde{f}$-local (in the sense of Definition 6.2.3.1).