$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.2.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pushout diagram
9.5
\begin{equation} \begin{gathered}\label{proposition:pushout-of-semiorthogonal} \xymatrix@C =50pt@R=50pt{ X \ar [d]^{w} \ar [r] & X' \ar [d]^{w'} \\ Y \ar [r] & Y'. } \end{gathered} \end{equation}
If an object $C \in \operatorname{\mathcal{C}}$ is weakly $w$-local, then it is also weakly $w'$-local.
Proof.
We have a commutative diagram of sets
\[ \xymatrix@C =30pt@R=50pt{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y', C ) \ar [r] & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X', C) \times _{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, C ) } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, C ) \ar [r] \ar [d] & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X', C) \ar [d] \\ & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y,C) \ar [r]^-{ \circ [w] } & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, C), } \]
where the square on the right is a pullback. Our assumption that $C$ is weakly $w$-local guarantees that the bottom horizontal map is surjective, so that the upper horizontal map on the right is also surjective. Since Proposition 9.5 is a pushout square, the horizontal map on the upper left is also surjective (Warning 7.6.2.3). It follows that the composite map $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y', C) \xrightarrow { \circ [w'] } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X',C)$ is also surjective.
$\square$