Theorem 9.6.3.3 (The Small Object Argument). Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$. For every object $X \in \operatorname{\mathcal{C}}$, there exists a morphism $f: X \rightarrow Y$ where $Y$ is weakly $W$-local and $f$ is a (small) transfinite pushout of morphisms of $W$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Theorem 9.6.3.3. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $W = \{ w_ s: C_ s \rightarrow D_ s \} $ be a small collection of morphisms of $\operatorname{\mathcal{C}}$. It follows from Remark 9.4.6.8 that there exists a small regular cardinal $\kappa $ such that each of the objects $C_ s$ is $\kappa $-compact. The desired result now follows from Proposition 9.6.3.9 (applied in the case where $\lambda = \Omega $ is a strongly inaccessible cardinal). $\square$