Lemma 8.7.3.8 (060W)

Lemma 8.7.3.8. Let $\mathbb {K}$ be a collection of simplicial sets, let $\lambda $ be an uncountable cardinal, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\lambda $-small $\infty $-categories. Suppose that $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocomplete, and that the $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small. Then the following conditions are equivalent:

$(1)$: The functor $F$ exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ (Definition 8.4.5.1).
$(2)$: For every $\infty $-category $\operatorname{\mathcal{E}}$ which is $\mathbb {K}$-cocomplete and $\lambda $-small, precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.
$(3)$: For every $\infty $-category $\operatorname{\mathcal{E}}$ which is $\mathbb {K}$-cocomplete and $\lambda $-small, precomposition with $F$ induces a homotopy equivalence of Kan complexes

\[ \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq }. \]
$(4)$: For every $\infty $-category $\operatorname{\mathcal{E}}$ which is $\mathbb {K}$-cocomplete and $\lambda $-small, precomposition with $F$ induces a bijection

\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{QC}}}_{< \lambda }^{ \mathbb {K}-\mathrm{cocont}} }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{QC}}}_{< \lambda } }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}). \]

Proof. The implications $(1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (4)$ are immediate. We complete the proof by showing that $(4) \Rightarrow (1)$. Choose a $\lambda $-small $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ and a functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. We may then assume without loss of generality that $F = \widehat{F} \circ h$, where $\widehat{F}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocontinuous. If condition $(4)$ is satisfied, then for every $\operatorname{\mathcal{E}}\in \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{cocont}}$, precomposition with the isomorphism class of the functor $\widehat{F}$ induces a bijection

\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{QC}}}_{< \lambda }^{ \mathbb {K}-\mathrm{cocont}} }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{QC}}}^{ \mathbb {K}-\mathrm{cocont}}_{< \lambda } }( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{E}}). \]

It follows that $[ \widehat{F} ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{QC}}}_{< \lambda }^{ \mathbb {K}-\mathrm{cocont}}$, so that $\widehat{F}$ is an equivalence of $\infty $-categories. Since $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$, it follows that $F = \widetilde{F} \circ h$ exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. $\square$