Corollary 9.2.7.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $S$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $g: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which is weakly right orthogonal to $S$. Assume that every morphism $f: A \rightarrow B$ which belongs to $S$ admits a relative codiagonal $\gamma _{A/B}: B \coprod _{A} B \rightarrow B$ which also belongs to $S$ (see Variant 7.6.2.16). Then $g$ is right orthogonal to $S$.
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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 $\pi ( \widetilde{X} ) = X$. Let $\widetilde{S}$ denote the collection of those morphisms $\widetilde{f}$ in $\operatorname{\mathcal{C}}_{/Y}$ which satisfy $\pi ( \widetilde{f} ) \in S$. Our assumption that $g$ is weakly right orthogonal to $S$ guarantees that $\widetilde{X}$ is weakly $\widetilde{S}$-local (Remark 9.2.5.12). It follows from Proposition 7.1.4.20 that every morphism of $\widetilde{S}$ admits a relative codiagonal which also belongs to $\widetilde{S}$, so that $\widetilde{X}$ is $\widetilde{S}$-local (Proposition 9.2.3.15). Invoking Corollary 9.2.7.13, we conclude that $g$ is right orthogonal to $S$. $\square$