Theorem 9.2.4.3 (The Small Object Argument). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Assume that:
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small and admits small colimits.
The collection $W$ is small.
For each morphism $w: X \rightarrow Y$ which belongs to $W$, the object $X \in \operatorname{\mathcal{C}}$ is $\kappa $-compact for some small cardinal $\kappa $ (see Definition 
).
For every object $C \in \operatorname{\mathcal{C}}$, there exists a morphism $f: C \rightarrow C'$ where $C'$ is weakly $W$-local and $f$ is a transfinite pushout of morphisms of $W$.
Proof of Theorem 9.2.4.3.
Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category which admits small colimits, let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\kappa $ be a small regular cardinal having the property that for each morphism $w: X \rightarrow Y$ which belongs to $W$, the object $X$ is $\kappa $-compact. Fix an object $C \in \operatorname{\mathcal{C}}$; we wish to show that there exists a morphism $f: C \rightarrow C'$ where $C'$ is weakly $W$-local and $f$ is a transfinite pushout of morphisms belonging to $W$. Without loss of generality, we may assume that $C$ itself is not weakly $W$-local (otherwise, we can take $f = \operatorname{id}_{C}$).
Let $W'$ be the collection of all morphisms of $\operatorname{\mathcal{C}}$ which are pushouts of morphisms of $W$, and let $\overline{W}$ denote the transfinite closure of $W'$ (Definition 9.2.2.13). Let $U \subseteq \overline{W}$ denote the subcollection consisting of those morphisms $u$ which satisfy the requirement of Lemma 9.2.4.8. Using Lemma 9.2.4.9 we deduce that there exists a diagram $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ where $F(0) = C$ and $F$ exhibits the induced map $F(0) \xrightarrow {f} F(\kappa )$ as a transfinite composition of morphisms of $U$.
We first claim that the object $C' = F(\kappa )$ is weakly $W$-local. Let $w: X \rightarrow Y$ be a morphism which belongs to $W$. We wish to show that every morphism $[ \overline{e} ]: X \rightarrow F(\kappa )$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ factors through the homotopy class $[w]$. Since $F$ is a colimit diagram and the object $X$ is $\kappa $-compact, the morphism $[ \overline{e} ]$ factors as a composition $X \xrightarrow { [e] } F(\alpha ) \rightarrow F(\kappa )$ for some ordinal $\alpha < \kappa $ and some morphism $e: X \rightarrow F(\alpha )$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Since the transition map $F( \alpha ) \rightarrow F(\alpha +1)$ belongs to $U$, we can choose a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [d] \ar [r]^-{w} \ar [d]^{e} & Y \ar [d]^{e'} \\ F(\alpha ) \ar [r] & F(\alpha +1). } \]
It follows that $[ \overline{e} ]$ factors as a composition $X \xrightarrow {[w]} Y \xrightarrow { [e'] } F(\alpha +1) \rightarrow F(\kappa )$.
To complete the proof, it will suffice to show that $f$ is a transfinite composition of morphisms belonging to $W'$. By construction, $f$ is a transfinite pushout of morphisms of $U \subseteq \overline{W}$, and therefore belongs to $\overline{W}$. This follows from Corollary 9.2.2.19, since $f$ is not an isomorphism (otherwise, the object $C$ would also be weakly $W$-local, contrary to our initial assumption).
$\square$