Lemma 9.3.2.5. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete, and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ be a full subcategory. Assume that $\operatorname{\mathcal{C}}$ generates $\operatorname{\mathcal{D}}$ under $\lambda $-small $\kappa $-filtered colimits. Then every $(\kappa ,\lambda )$-compact object $X \in \operatorname{\mathcal{D}}$ is a retract of an object of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\operatorname{\mathcal{D}}' \subseteq \operatorname{\mathcal{D}}$ be the full subcategory spanned by those objects $Y$ which satisfy the following condition:
- $(\ast )$
Every morphism $f: X \rightarrow Y$ factors (up to homotopy) through an object of $\operatorname{\mathcal{C}}$.
Then $\operatorname{\mathcal{D}}'$ contains $\operatorname{\mathcal{C}}$, and our assumption that $X$ is $(\kappa ,\lambda )$-compact guarantees that $\operatorname{\mathcal{D}}'$ is closed under the formation of $\lambda $-small $\kappa $-filtered colimits. Since $\operatorname{\mathcal{C}}$ generates $\operatorname{\mathcal{D}}$ under $\lambda $-small $\kappa $-filtered colimits, we must have $\operatorname{\mathcal{D}}' = \operatorname{\mathcal{D}}$. In particular, $\operatorname{\mathcal{D}}'$ contains the object $X$, so the identity morphism $\operatorname{id}_{X}: X \rightarrow X$ factors through an object of $\operatorname{\mathcal{C}}$. $\square$