Remark 9.2.1.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pushouts, let $w: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $\gamma _{X/Y}: Y \coprod _{X} Y \rightarrow Y$ be the relative codiagonal of $w$ (see Variant 7.6.2.16). If an object $C \in \operatorname{\mathcal{C}}$ is $w$-local, then it is also $\gamma _{X/Y}$-local. This follows by applying Remark 9.2.1.11 to the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \coprod _{X} Y \ar [dr]^{\gamma _{X/Y}} & \\ Y \ar [ur]^{w'} \ar [rr]^{\operatorname{id}} & & Y; } \]
here $w'$ is a pushout of $w$ (so that $C$ is $w'$-local by virtue of Remark 9.2.1.10). For a partial converse, see Exercise 9.2.3.16.