Corollary 9.2.7.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, so that $g$ can be identified with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. The following conditions are equivalent:
- $(1)$
The morphism $g$ is right orthogonal to $f$ (in the sense of Definition 9.2.7.4).
- $(2)$
For every morphism $\widetilde{f}: \widetilde{A} \rightarrow \widetilde{B}$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f} ) = f$, the object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ is $\widetilde{f}$-local (in the sense of Definition 9.2.1.1).