Remark 9.2.5.12 (04N2)

Remark 9.2.5.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a morphism $g: X \rightarrow Y$, which we identify with an object $\widetilde{X}$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. Let $S$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\widetilde{S}$ denote its inverse image in $\operatorname{\mathcal{C}}_{/Y}$. The following conditions are equivalent:

The morphism $g$ is weakly right orthogonal to $S$ (in the sense of Variant 9.2.5.11).
The object $\widetilde{X}$ is weakly $\widetilde{S}$-local (in the sense of Definition 9.2.3.5).