$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.5.5.5. Let $\kappa \trianglelefteq \lambda $ be regular cardinals and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{C}}$. Then the composite functor
\[ \operatorname{\mathcal{C}}\xrightarrow {h} \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} ) \]
exhibits $\operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$ as a $\lambda $-cocompletion of $\operatorname{\mathcal{C}}$.
Proof.
Using Proposition 8.4.5.3, we can choose a functor $h': \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}'$ which exhibits $\widehat{\operatorname{\mathcal{C}}}'$ as a $\lambda $-cocompletion of $\operatorname{\mathcal{C}}$. Without loss of generality, we may assume that $\widehat{\operatorname{\mathcal{C}}}$ is the smallest full subcategory of $\widehat{\operatorname{\mathcal{C}}}'$ which contains the essential image of $h'$ and is closed under $\kappa $-small colimits. Let $T: \operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} ) \rightarrow \widehat{\operatorname{\mathcal{C}}}'$ be an $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of the inclusion functor $\widehat{\operatorname{\mathcal{C}}} \hookrightarrow \widehat{\operatorname{\mathcal{C}}}'$ (Definition 9.3.1.12). It follows from Proposition 9.2.5.24 that every object of $\widehat{\operatorname{\mathcal{C}}}$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\widehat{\operatorname{\mathcal{C}}}'$, so the functor $T$ is fully faithful (Proposition 9.3.2.1). To complete the proof, it will suffice to show that $T$ is essentially surjective. Without loss of generality, we may assume that $\lambda $ is uncountable (otherwise, the result is immediate from Example 9.3.1.10). In this case, Variant 9.3.4.18 guarantees that every object $X \in \widehat{\operatorname{\mathcal{C}}}'$ can be realized as the colimit of a diagram $K \xrightarrow {F} \operatorname{\mathcal{C}}\xrightarrow {h'} \widehat{\operatorname{\mathcal{C}}}'$, where $K$ is a $\lambda $-small simplicial set. Applying Lemma 9.1.7.18 we can realize $K$ as the colimit of a diagram
\[ A \rightarrow \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto K_{\alpha }, \]
where $A$ is a $\lambda $-small $\kappa $-directed partially ordered set and each $K_{\alpha }$ is a $\kappa $-small simplicial set. For each $\alpha \in A$, the diagram $(h' \circ F)|_{ K_{\alpha } }$ has some colimit $X_{\alpha } \in \widehat{\operatorname{\mathcal{C}}}$. Since $K$ is also a categorical colimit of the diagram $\{ K_{\alpha } \} _{\alpha \in A}$ (Proposition 9.1.6.1), we can promote the construction $\alpha \mapsto X_{\alpha }$ to a $\lambda $-small $\kappa $-filtered diagram $\operatorname{N}_{\bullet }(A) \rightarrow \widehat{\operatorname{\mathcal{C}}}$ having colimit $X$ (see Proposition 7.5.8.12), so that $X$ belongs to the essential image of $T$.
$\square$