Lemma 9.6.3.7 (04MF)

Lemma 9.6.3.7. Let $\lambda $ be a regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\lambda $-cocomplete, and let $W$ be a $\lambda $-small collection of morphisms of $\operatorname{\mathcal{C}}$. Assume that, for every morphism $w: C \rightarrow D$ of $W$ and every object $X \in \operatorname{\mathcal{C}}$, the set of homotopy classes $\operatorname{Hom}_{\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}}( C, X )$ is $\lambda $-small. Then, for every object $X \in \operatorname{\mathcal{C}}$, there exists a morphism $f: X \rightarrow Y$ which is a $\lambda $-small transfinite pushout of morphisms of $W$ having the following property: for every morphism $w: C \rightarrow D$ which belongs to $W$ and every morphism $e: C \rightarrow X$, there exists a commutative diagram

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

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

Proof. For each morphism $w: C \rightarrow D$ belonging to $W$, let $\{ e_{s}: C \rightarrow X \} _{ s \in S_{w} }$ be a set of representatives for the homotopy classes of morphisms from $C$ to $X$. Since $\lambda $ is regular, the disjoint union $S = \coprod _{w \in W} S_ w$ is small. The desired result now follows by applying Lemma 9.6.3.6 to the collection $\{ e_ s \} _{s \in S}$. $\square$