Corollary 9.6.6.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: A \rightarrow B$ be a morphism in $\operatorname{\mathcal{C}}$ which admits a relative codiagonal $\gamma _{A/B}: B \coprod _{A} B \rightarrow B$ (see Variant 7.6.2.23). If $g$ is a morphism in $\operatorname{\mathcal{C}}$ which is right orthogonal to $f$, then it is right orthogonal to $\gamma _{A/B}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ & B \coprod _{A} B \ar [dr]^{\gamma _{A/B}} & \\ B \ar [ur]^{f'} \ar [rr]^{\operatorname{id}_ B} & & B, } \]
where $f'$ is a pushout of $f$ and therefore left orthogonal to $g$ (Corollary 9.6.6.14). Since $\operatorname{id}_{B}$ is automatically left orthogonal to $g$ (Remark 9.6.6.9), it follows that $\gamma _{A/B}$ is also left orthogonal to $g$ (Proposition 9.6.6.10). $\square$