Lemma 9.2.4.8 (04MF)

Lemma 9.2.4.8. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category which admits small colimits, and let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$. For every object $C \in \operatorname{\mathcal{C}}$, there exists a morphism $u: C \rightarrow D$ which is a transfinite pushout of morphisms of $W$ with the following property: for every morphism $w: X \rightarrow Y$ which belongs to $W$ and every morphism $e: X \rightarrow C$, there exists a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{w} \ar [d]^{e} & Y \ar [d] \\ C \ar [r]^-{u} & D } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$.

Proof. For each morphism $w: X \rightarrow Y$ belonging to $W$, let $\{ f_{s}: X \rightarrow C\} _{ s \in S_{w} }$ be a set of representatives for the homotopy classes of morphisms from $X$ to $C$. Since $\operatorname{\mathcal{C}}$ is locally small, the collection $S_{w}$ is small. Our assumption that $W$ is small then guarantees that the disjoint union $S = \coprod _{w \in W} S_{w}$ is small. The desired result now follows by applying Lemma 9.2.4.7 to the collection of morphisms $\{ f_ s \} _{s \in S}$. $\square$