8.7.3 Functoriality of Cocompletion
Let $\mathbb {K}$ be a collection of simplicial sets. It follows from Proposition 8.4.5.3 that every $\infty $-category $\operatorname{\mathcal{C}}$ admits a $\mathbb {K}$-cocompletion $\widehat{\operatorname{\mathcal{C}}}$. Our goal in this section is to upgrade the construction $\operatorname{\mathcal{C}}\mapsto \widehat{\operatorname{\mathcal{C}}}$ to a functor of $\infty $-categories. We begin by articulating the properties that we would like this functor to have.
Definition 8.7.3.1. Let $\mathbb {K}$ be a collection of simplicial sets. We say that a functor $T: \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ is a $\mathbb {K}$-cocompletion functor if there exists a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{QC}}} \rightarrow T$ satisfying the following conditions:
For every small $\infty $-category $\operatorname{\mathcal{C}}$, the functor $\eta _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow T(\operatorname{\mathcal{C}})$ exhibits $T(\operatorname{\mathcal{C}})$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$.
For every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ between small $\infty $-categories, the functor $T(F): T(\operatorname{\mathcal{C}}) \rightarrow T(\operatorname{\mathcal{D}})$ is $\mathbb {K}$-cocontinuous.
If these conditions are satisfied, we say that $\eta $ exhibits $T$ as a $\mathbb {K}$-cocompletion functor.
We can state a preliminary version of our main result as follows:
Proposition 8.7.3.2 (Functoriality of Cocompletion). Let $\mathbb {K}$ be a small collection of small simplicial sets. Then there exists a $\mathbb {K}$-cocompletion functor $T: \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$, which is uniquely determined up to isomorphism.
We will deduce Proposition 8.7.3.2 from a more precise result, which allows us to control the sizes of the cocompletions under consideration.
Definition 8.7.3.3. Let $\mathbb {K}$ be a collection of simplicial sets and let $\lambda $ be an uncountable cardinal. We say that a functor $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ is a $\mathbb {K}$-cocompletion functor if there exists a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{QC}}} \rightarrow T$ satisfying the following conditions:
For every $\lambda $-small $\infty $-category $\operatorname{\mathcal{C}}$, the functor $\eta _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow T(\operatorname{\mathcal{C}})$ exhibits $T(\operatorname{\mathcal{C}})$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$.
For every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ between $\lambda $-small $\infty $-categories, the functor $T(F): T(\operatorname{\mathcal{C}}) \rightarrow T(\operatorname{\mathcal{D}})$ is $\mathbb {K}$-cocontinuous.
If these conditions are satisfied, we say that $\eta $ exhibits $T$ as a $\mathbb {K}$-cocompletion functor.
In the special case where $\mathbb {K}$ is the collection of all small simplicial sets, we say that $T$ is a cocompletion functor if it is $\mathbb {K}$-cocompletion functor. More generally, if $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets (for some infinite cardinal $\kappa $), we say that $T$ is a $\kappa $-cocompletion functor if it is a $\mathbb {K}$-cocompletion functor.
Proposition 8.7.3.2 can be regarded as a special case of the following result, which we prove at the end of this section:
Proposition 8.7.3.5. Let $\kappa < \lambda $ be infinite cardinals, where $\lambda $ is regular and exponential cofinality $\geq \kappa $, and let $\mathbb {K}$ be a collection of $\kappa $-small simplicial sets. Then there exists a $\mathbb {K}$-cocompletion functor $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$, which is uniquely determined up to isomorphism.
Example 8.7.3.6. Following the convention of Remark 4.7.0.5, we say that a simplicial set is small if it is $\operatorname{\textnormal{\cjRL {t}}}$-small, for some fixed strongly inaccessible cardinal $\operatorname{\textnormal{\cjRL {t}}}$. Applying Proposition 8.7.3.5 in the special case $\kappa = \operatorname{\textnormal{\cjRL {t}}}$ and $\lambda = \operatorname{\textnormal{\cjRL {t}}}^{+}$, we deduce that there exists a cocompletion functor $T: \operatorname{\mathcal{QC}}_{< \operatorname{\textnormal{\cjRL {t}}}^{+} } \rightarrow \operatorname{\mathcal{QC}}_{< \operatorname{\textnormal{\cjRL {t}}}^{+} }$. The restriction of $T$ to small $\infty $-categories can be described more informally as follows:
For every small $\infty $-category $\operatorname{\mathcal{C}}$, the functor $T$ assigns the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (see Theorem 8.4.0.3).
To every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ between small $\infty $-categories, $T(F): \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is left adjoint to the functor given by precomposition with $F^{\operatorname{op}}$ (see Remark 8.7.3.4 and Example 8.4.4.5).
To prove Proposition 8.7.3.5, it will be convenient to work with a reformulation of Definition 8.7.3.3.
Notation 8.7.3.7. Let $\lambda $ be an uncountable cardinal, and let $\operatorname{\mathcal{QC}}_{< \lambda }$ denote the $\infty $-category of $\lambda $-small $\infty $-categories (Variant 5.5.4.11). For every collection of simplicial sets $\mathbb {K}$, we define a subcategory $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{cocont}} \subseteq \operatorname{\mathcal{QC}}_{< \lambda }$ as follows:
An object $\operatorname{\mathcal{C}}\in \operatorname{\mathcal{QC}}_{< \lambda }$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda }$ if and only if $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete: that is, it admits $K$-indexed colimits for each $K \in \mathbb {K}$
A morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ of $\operatorname{\mathcal{QC}}_{< \lambda }$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda }$ if and only if it is $\mathbb {K}$-cocontinuous: that is, it preserves $K$-indexed colimits for each $K \in \mathbb {K}$.
We will be particularly interested in the special case where $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets, for some infinite cardinal $\kappa $. In this case, we denote the $\infty $-category $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda }$ by $\operatorname{\mathcal{QC}}^{\kappa -\mathrm{cocont}}_{< \lambda }$.
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 $\operatorname{\mathcal{C}}$.
$\square$
Proposition 8.7.3.9. Let $\mathbb {K}$ be a collection of simplicial sets and let $\lambda $ be an uncountable cardinal. Assume that every $\lambda $-small $\infty $-category $\operatorname{\mathcal{C}}$ admits a $\lambda $-small $\mathbb {K}$-cocompletion. Then:
- $(1)$
The inclusion functor $\iota : \operatorname{\mathcal{QC}}_{<\lambda }^{\mathbb {K}-\mathrm{cocont}} \hookrightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ admits a left adjoint.
- $(2)$
If $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{<\lambda }^{\mathbb {K}-\mathrm{cocont}}$ is a functor, then a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{QC}}_{< \lambda } } \rightarrow \iota \circ T$ is the unit of an adjunction between $T$ and $\iota $ (in the sense of Definition 6.2.1.1) if and only if it exhibits $T$ as a $\mathbb {K}$-cocompletion functor (in the sense of Definition 8.7.3.3).
Proof.
Assertion $(1)$ follows by combining Lemma 8.7.3.8 with Proposition 6.2.6.1. Assertion $(2)$ follows by combining Lemma 8.7.3.8 with Corollary 6.2.6.5.
$\square$
Corollary 8.7.3.10. Let $\mathbb {K}$ be a collection of simplicial sets and let $\lambda $ be an uncountable cardinal. The following conditions are equivalent:
- $(1)$
Every $\lambda $-small $\infty $-category $\operatorname{\mathcal{C}}$ admits a $\mathbb {K}$-cocompletion which is also $\lambda $-small.
- $(2)$
There exists a $\mathbb {K}$-cocompletion functor $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{<\lambda }$.
Moreover, if these conditions are satisfied, then a functor $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ is a $\mathbb {K}$-cocompletion functor if and only if it factors through the subcategory $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{cocont}}$ and is left adjoint to the inclusion functor $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{cocont}} \hookrightarrow \operatorname{\mathcal{QC}}_{< \lambda }$.
To deduce Proposition 8.7.3.5 from Corollary 8.7.3.10, we need to control the sizes of $\mathbb {K}$-cocompletions.
Proposition 8.7.3.11. Let $\lambda $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\lambda $-small. Let $\mathbb {K}$ be a collection of $\kappa $-small simplicial sets, where $\kappa = \mathrm{ecf}(\lambda )$ is the exponential cofinality of $\lambda $. Then the $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ is also locally $\lambda $-small.
Proof.
Choose a functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Fix objects $\widehat{X},\widehat{Y} \in \widehat{\operatorname{\mathcal{C}}}$; we wish to show that the morphism space $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \widehat{X}, \widehat{Y})$ is essentially $\lambda $-small. Let us first regard the object $\widehat{Y}$ as fixed, and let $\widehat{\operatorname{\mathcal{C}}}_0$ denote the full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ spanned by those objects $\widehat{X}$ for which the morphism space $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \widehat{X}, \widehat{Y})$ is essentially $\lambda $-small. Since $\lambda $ has exponential cofinality $\geq \kappa $, the collection of $\lambda $-small spaces is closed under formation of $\kappa $-small limits (Variant 7.4.1.4). It follows that $\widehat{\operatorname{\mathcal{C}}}$ is closed under $\kappa $-small colimits in $\widehat{\operatorname{\mathcal{C}}}$ (see Proposition 7.4.1.18). In particular, it is closed under $K$-indexed colimits for each $K \in \mathbb {K}$. Consequently, to show that $\widehat{\operatorname{\mathcal{C}}}_0 = \widehat{\operatorname{\mathcal{C}}}$, it will suffice to show that the mapping space $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \widehat{X}, \widehat{Y})$ is essentially $\lambda $-small in the special case $\widehat{X} = h(X)$ for some object $X \in \operatorname{\mathcal{C}}$.
Let us now regard the object $\widehat{X} = h(X)$ as fixed, and let $\widehat{\operatorname{\mathcal{C}}}_1$ denote the full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ spanned by those objects $\widehat{Y}$ for which the morphism space $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \widehat{X}, \widehat{Y})$ is essentially $\lambda $-small. For each $K \in \mathbb {K}$, the $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( h(X), \bullet )$ preserves $\mathbb {K}$-indexed colimits. Since the collection of $\lambda $-small spaces is closed under $K$-indexed colimits (Corollary 7.4.3.8), it follows that the subcategory $\widehat{\operatorname{\mathcal{C}}}_{1} \subseteq \widehat{\operatorname{\mathcal{C}}}$ is closed under $K$-indexed colimits. Consequently, to prove that $\widehat{\operatorname{\mathcal{C}}}_{1} = \widehat{\operatorname{\mathcal{C}}}$, it will suffice to show that it contains every object of the form $h(Y)$, for $Y \in \operatorname{\mathcal{C}}$. Since $h$ is fully faithful, we are reduced to proving that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is essentially $\lambda $-small, which follows from our assumption that $\operatorname{\mathcal{C}}$ is locally $\lambda $-small.
$\square$
Example 8.7.3.12. Let $\kappa $ be a regular cardinal and let $\lambda $ be an uncountable cardinal of exponential cofinality $\geq \kappa $. If $\operatorname{\mathcal{C}}$ is a locally $\lambda $-small $\infty $-category, then the $\kappa $-cocompletion of $\operatorname{\mathcal{C}}$ is also locally $\lambda $-small.
Corollary 8.7.3.13. Let $\mathbb {K}$ be a collection of small simplicial sets and let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. Then the $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ is also locally small.
Proof.
Apply Proposition 8.7.3.11 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a strongly inaccessible cardinal (see Remark 8.7.3.11).
$\square$
Example 8.7.3.14. If $\operatorname{\mathcal{C}}$ is a locally small $\infty $-category, then the cocompletion of $\operatorname{\mathcal{C}}$ is also locally small.
Proposition 8.7.3.15. Let $\kappa < \lambda $ be infinite cardinals, where $\lambda $ is regular and has exponential cofinality $\geq \kappa $. Let $\mathbb {K}$ be a collection of $\kappa $-small simplicial sets. If $\operatorname{\mathcal{C}}$ is an essentially $\lambda $-small $\infty $-category, then its $\mathbb {K}$-cocompletion is essentially $\lambda $-small.
Proof of Proposition 8.7.3.15.
Let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibit $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Proposition 8.7.3.11 guarantees that the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is locally $\lambda $-small. It will therefore suffice to show that the set of isomorphism classes $\pi _0( \widehat{\operatorname{\mathcal{C}}}^{\simeq } )$ is $\lambda $-small (Proposition 4.7.8.7).
We define a transfinite sequence of full subcategories $\{ \widehat{\operatorname{\mathcal{C}}}_{\beta } \subseteq \widehat{\operatorname{\mathcal{C}}} \} _{\beta \leq \kappa }$ as follows:
We define $\widehat{\operatorname{\mathcal{C}}}_{0}$ to be the essential image of the functor $h$.
If $\beta = \alpha +1$ is a successor ordinal, we define $\widehat{\operatorname{\mathcal{C}}}_{\beta }$ to be the smallest replete full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ which contains $\widehat{\operatorname{\mathcal{C}}}_{\alpha }$ and the colimit of every diagram $K \rightarrow \widehat{\operatorname{\mathcal{C}}}_{\alpha |}$ for $K \in \mathbb {K}$.
If $\beta \leq \lambda $ is a nonzero limit ordinal, we set $\widehat{\operatorname{\mathcal{C}}}_{\beta } = \bigcup _{\alpha < \beta } \widehat{\operatorname{\mathcal{C}}}_{\alpha }$.
By assumption, every $K \in \mathbb {K}$ is $\kappa $-small. Consequently, every diagram $K \rightarrow \widehat{\operatorname{\mathcal{C}}}_{\kappa }$ factors through $\widehat{\operatorname{\mathcal{C}}}_{\alpha }$ for some $\alpha < \kappa $, and therefore has a colimit in $\widehat{\operatorname{\mathcal{C}}}_{\alpha +1} \subseteq \widehat{\operatorname{\mathcal{C}}}_{\kappa }$. It follows that $\widehat{\operatorname{\mathcal{C}}}_{\kappa }$ contains the essential image of $h$ and is closed under $K$-indexed colimits for each $K \in \mathbb {K}$, and therefore coincides with $\widehat{\operatorname{\mathcal{C}}}$. We are therefore reduced to showing that the set of isomorphism classes $\pi _0( \widehat{\operatorname{\mathcal{C}}}_{\kappa }^{\simeq } )$ is $\lambda $-small. Suppose otherwise. Then there is some smallest ordinal $\beta \leq \kappa $ such that $\pi _0( \widehat{\operatorname{\mathcal{C}}}_{\beta }^{\simeq } )$ is not $\lambda $-small. Our assumption that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small guarantees that $\beta > 0$. Since $\lambda $ has cofinality $> \kappa $, $\beta $ cannot be a limit ordinal. We must therefore have $\beta = \alpha +1$, for some ordinal $\alpha < \kappa $. In this case, the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}_{\alpha }$ is essentially $\mu $-small.
Note that every object of $\widehat{\operatorname{\mathcal{C}}}_{\alpha +1}$ either belongs to $\widehat{\operatorname{\mathcal{C}}}_{\alpha }$ or can be realized as the colimit of a diagram $F: K \rightarrow \widehat{\operatorname{\mathcal{C}}}_{\alpha }$, for some $K \in \mathbb {K}$. Note that we may assume that $\mathbb {K}$ is $\lambda $-small (Remark 8.7.3.16). Moreover, if $K \in \mathbb {K}$ is fixed, our assumption that $\lambda $ has exponential cofinality $\geq \kappa $ guarantees that the set of isomorphism classes $\pi _0( \operatorname{Fun}( K, \widehat{\operatorname{\mathcal{C}}}_{\alpha })^{\simeq } )$ is $\lambda $-small (Corollary 4.7.8.8). The regularity of $\lambda $ now guarantees that $\pi _0( \widehat{\operatorname{\mathcal{C}}}_{\alpha +1}^{\simeq } )$ is $\lambda $-small, contradicting our choice of $\beta $.
$\square$
Example 8.7.3.17. Let $\kappa < \lambda $ be infinite cardinals, where $\lambda $ is regular and of exponential cofinality $\geq \kappa $. If an $\infty $-category $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, then its $\kappa $-cocompletion is also essentially $\lambda $-small.
Example 8.7.3.18. Let $\mathbb {K}$ be a small collection of small simplicial sets. If $\operatorname{\mathcal{C}}$ is an essentially small $\infty $-category, then its $\mathbb {K}$-cocompletion is also essentially small. Beware that the smallness condition on $\mathbb {K}$ cannot be omitted: for example, the cocompletion of a (nonempty) $\infty $-category $\operatorname{\mathcal{C}}$ is never essentially small (see Proposition 7.1.2.15).
Proof of Proposition 8.7.3.5.
Combine Proposition 8.7.3.15 with Corollary 8.7.3.10.
$\square$