Proposition 9.3.3.1. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Suppose that $\operatorname{\mathcal{C}}$ is locally $\mu $-small, where $\mu \geq \lambda $ is a regular cardinal of exponential cofinality $\geq \lambda $. Then $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is also locally $\mu $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.3.3 Functoriality of Ind-Completion
In ยง9.3.1, we observed that every $\infty $-category $\operatorname{\mathcal{C}}$ admits an $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}})$, which is well-defined up to equivalence (Notation 9.3.1.2). In this section, we show that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}(\operatorname{\mathcal{C}})$ can be upgraded to a functor of $\infty $-categories. To formulate this result precisely, we will need to take some care about the sizes of the $\infty $-categories under consideration.
Proof. This is a special case of Proposition 8.4.6.10. $\square$
Corollary 9.3.3.2. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. Then the $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}})$ is also locally small.
Proof. Apply Proposition 9.3.3.1 in the special case where $\kappa = \aleph _0$ and $\lambda = \mu $ is a strongly inaccessible cardinal. $\square$
Variant 9.3.3.3. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Suppose that $\operatorname{\mathcal{C}}$ is essentially $\mu $-small, where $\mu > \lambda $ is a regular cardinal of exponential cofinality $\geq \lambda $. Then $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is also essentially $\mu $-small.
Proof. This is a special case of Proposition 8.4.6.14. $\square$
Warning 9.3.3.4. In contrast with Proposition 9.3.3.1, the conclusion of Variant 9.3.3.3 generally does not hold in the limiting case $\lambda = \mu $. For example, if $\operatorname{\mathcal{C}}$ is an essentially small $\infty $-category, then $\operatorname{Ind}(\operatorname{\mathcal{C}})$ is usually not essentially small.
Definition 9.3.3.5. Let $\kappa \leq \lambda \leq \mu $ be regular cardinals, where $\mu $ is uncountable. We say that a functor $T: \operatorname{\mathcal{QC}}_{< \mu } \rightarrow \operatorname{\mathcal{QC}}_{< \mu }$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion functor if there exists a natural transformation $\eta : \operatorname{id}_{ \operatorname{\mathcal{QC}}_{< \mu } } \rightarrow T$ satisfying the following conditions:
For every $\mu $-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 an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$.
For every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ between $\mu $-small $\infty $-categories, the functor $T(F): T(\operatorname{\mathcal{C}}) \rightarrow T(\operatorname{\mathcal{D}})$ is $(\kappa ,\lambda )$-finitary.
If these conditions are satisfied, we say that $\eta $ exhibits $T$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion functor.
Remark 9.3.3.6. Let $\kappa \leq \lambda $ be regular cardinals, and let $\mathbb {K}$ be the collection of all $\lambda $-small $\kappa $-filtered $\infty $-categories. Then $T: \operatorname{\mathcal{QC}}_{< \mu } \rightarrow \operatorname{\mathcal{QC}}_{< \mu }$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion functor (in the sense of Definition 9.3.3.5) if and only if it is a $\mathbb {K}$-cocompletion functor (in the sense of Definition 8.4.6.3).
Proposition 9.3.3.7. Let $\kappa \leq \lambda < \mu $ be regular cardinals. If $\mu $ has exponential cofinality $\geq \lambda $, then there exists an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion functor $T: \operatorname{\mathcal{QC}}_{< \mu } \rightarrow \operatorname{\mathcal{QC}}_{< \mu }$. Moreover, $T$ is uniquely determined up to isomorphism.
Proof. By virtue Corollary 8.4.6.9 (and Remark 9.3.3.6), it will suffice to observe that if $\operatorname{\mathcal{C}}$ an essentially $\mu $-small $\infty $-category, then $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is also essentially $\mu $-small, which follows from Variant 9.3.3.3. $\square$
Notation 9.3.3.8. Let $\kappa \leq \lambda < \mu $ be regular cardinals, where $\mu $ has exponential cofinality $\geq \lambda $. Proposition 9.3.3.7 asserts that there exists an essentially unique $\operatorname{Ind}_{\kappa }^{\lambda }$-completion functor $T: \operatorname{\mathcal{QC}}_{< \mu } \rightarrow \operatorname{\mathcal{QC}}_{< \mu }$. To emphasize its uniqueness, we will typically denote the functor $T(-)$ by $\operatorname{Ind}_{\kappa }^{\lambda }(-)$ (compare with Notation 9.3.1.8). Following the convention of Remark 4.7.0.5, we say that $\kappa $ is small if it satisfies $\kappa < \operatorname{\Omega }$, for some fixed strongly inaccessible cardinal $\operatorname{\Omega }$. In this case, we typically denote the functor $\operatorname{Ind}_{\kappa }^{\operatorname{\Omega }}(-)$ by $\operatorname{Ind}_{\kappa }(-)$. In the special case where $\kappa = \aleph _0$, we also denote the functor $\operatorname{Ind}_{\kappa }(-)$ by $\operatorname{Ind}(-)$.
Remark 9.3.3.9. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Applying the functor $\operatorname{Ind}_{\kappa }^{\lambda }(-)$ of Notation 9.3.3.8 (for any sufficiently large $\mu $), we obtain a $(\kappa ,\lambda )$-finitary functor $\operatorname{Ind}_{\kappa }^{\lambda }(F): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$, for which the diagram commutes up to isomorphism. It follows that $\operatorname{Ind}_{\kappa }^{\lambda }(F)$ agrees (up to isomorphism) with the functor defined in Notation 9.3.1.13, given by the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of the composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r] \ar [d]^{F} & \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{Ind}_{\kappa }^{\lambda }(F) } \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}}) } \]
Remark 9.3.3.10. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Choose a regular cardinal $\mu \geq \lambda $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\mu $-small, and let be covariant Yoneda embeddings for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$. By virtue of Example 8.4.4.5, left Kan extension along $F$ determines a $\mu $-cocontinuous functor $F_{!}: \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{ < \mu } )$ for which the diagram commutes up to (canonical) isomorphism. Using Proposition 8.4.5.8, we can identify $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ and $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ with the smallest full subcategories of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ and $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ which are closed under $\lambda $-small $\kappa $-filtered colimits and which contain the essential images of $h_{\bullet }^{\operatorname{\mathcal{C}}}$ and $h_{\bullet }^{\operatorname{\mathcal{D}}}$, respectively. It follows from the commutativity of the diagram (9.13) that $F_{!}$ restricts to a functor $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$, which we can identify with the functor $\operatorname{Ind}_{\kappa }^{\lambda }(F)$ of Remark 9.3.3.9.
\[ h_{\bullet }^{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) \quad \quad h_{\bullet }^{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) \]
9.13
\begin{equation} \begin{gathered}\label{equation:Ind-functor-via-LKE} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{F} \ar [r]^-{ h_{\bullet }^{\operatorname{\mathcal{C}}} } & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) \ar [d]^{F_{!}} \\ \operatorname{\mathcal{D}}\ar [r]^-{ h_{\bullet }^{\operatorname{\mathcal{D}}} } & \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) } \end{gathered} \end{equation}
Exercise 9.3.3.11. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Show that $\operatorname{Ind}_{\kappa }^{\lambda }(G): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is right adjoint to $\operatorname{Ind}_{\kappa }^{\lambda }(F): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$.
Proposition 9.3.3.12. Let $\kappa \leq \lambda < \mu $ be regular cardinals, where $\mu $ has exponential cofinality $\geq \lambda $. Then the functor is $(\lambda , \mu )$-finitary: that is, it commutes with $\mu $-small $\lambda $-filtered colimits.
\[ \operatorname{Ind}_{\kappa }^{\lambda }: \operatorname{\mathcal{QC}}_{< \mu } \rightarrow \operatorname{Cat}{< \mu } \]
Proof. By virtue of Remark 9.3.3.6, this is a special case of Corollary 9.1.9.11. $\square$
We now study a somewhat different way of articulating the functoriality of $\operatorname{Ind}$-completion: it is a construction which makes sense in families.
Definition 9.3.3.13. Let $\kappa \leq \lambda $ be regular cardinals and suppose we are given a commutative diagram of simplicial sets where $U$ and $\widehat{U}$ are inner fibrations. We will say that $H$ exhibits $\widehat{U}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $U$ if the following condition is satisfied:
For every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the functor
exhibits the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, in the sense of Variant 9.3.1.7.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{U} \ar [rr]^{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \]
\[ H_{\sigma }: \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}} \]
Warning 9.3.3.14. The terminology of Definition 9.3.3.13 is potentially confusing. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. We have now assigned two different meanings to the term $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $U$:
The $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $U$ as a functor (in the sense of Remark 9.3.3.9): this is a functor $\operatorname{Ind}_{\kappa }^{\lambda }(U): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$.
The $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $U$ as an inner fibration (in the sense of Definition 9.3.3.13): this is an inner fibration $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$.
Beware that these are distinct notions. However, they are closely related: one can arrange that they fit into a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \widehat{\operatorname{\mathcal{E}}} \ar [d]^{ \widehat{U} } \ar [r] & \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{E}}) \ar [d]^{ \operatorname{Ind}_{\kappa }^{\lambda }(U)} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) } \]
which is often (but not always) a categorical pullback square. See Proposition 
.
Proposition 9.3.3.15. Let $\kappa \leq \lambda $ be regular cardinals and suppose we are given a commutative diagram of simplicial sets where $U$ and $\widehat{U}$ are inner fibrations. Then $H$ exhibits $\widehat{U}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $U$ if and only if the following pair of conditions is satisfied:
The inner fibration $\widehat{U}$ is $(\kappa ,\lambda )$-cocomplete (in the sense of Definition 9.2.9.4).
For each vertex $C \in \operatorname{\mathcal{C}}$, the map of fibers $H_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{C}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{E}}_{C}$ (in the sense of Variant 9.3.1.7).
For every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$ and every object $X \in \operatorname{\mathcal{E}}_{C}$, the image $H(X)$ is $(\kappa ,\lambda )$-compact as an object of the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{U} \ar [rr]^{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \]
Proof. If $H$ exhibits $\widehat{U}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $U$, then conditions $(1)$ and $(2)$ are immediate from the definitions and $(3)$ follows from the recognition principle of Proposition 9.3.2.3. We now prove the converse. Assume that conditions $(1)$, $(2)$, and $(3)$ are satisfied and let $\sigma $ be an $n$-simplex of $\operatorname{\mathcal{C}}$; we wish to show that the induced map $H_{\sigma }: \operatorname{\mathcal{E}}_{\sigma } \rightarrow \widehat{\operatorname{\mathcal{E}}}_{\sigma }$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{\sigma }$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{E}}_{\sigma }$. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex and that $\sigma $ is the identity map; in this case, we wish to show that $H$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{E}}$. It follows from assumption $(1)$ that $\widehat{\operatorname{\mathcal{E}}}$ is $(\kappa ,\lambda )$-cocomplete. Moreover, for $0 \leq i \leq n$, the full subcategory $\widehat{\operatorname{\mathcal{E}}}_{i} = \widehat{U}^{-1} \{ i\} $ is closed under the formation of $\lambda $-small $\kappa $-filtered colimits in $\widehat{\operatorname{\mathcal{E}}}_ i$. Assumption $(2)$ guarantees that each $\widehat{\operatorname{\mathcal{E}}}_{i}$ is generated under $\lambda $-small $\kappa $-filtered colimits by the essential image of $\operatorname{\mathcal{E}}_ i$. To complete the proof, it will suffice to show that for each object $X \in \operatorname{\mathcal{E}}$, the image $H(X) \in \widehat{\operatorname{\mathcal{E}}}$ is $(\kappa ,\lambda )$-compact (see Proposition 9.3.2.3). By virtue of Remark 9.2.9.13, this is a reformulation of assumption $(3)$. $\square$
Remark 9.3.3.16. Let $\kappa \leq \lambda $ be regular cardinals, let $\mathbb {K}$ be the collection of all $\lambda $-small $\kappa $-filtered $\infty $-categories, and suppose we are given a commutative diagram of simplicial sets where $U$ and $\widehat{U}$ are inner fibrations. Proposition 9.3.3.15 asserts that the following conditions are equivalent:
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{U} \ar [rr]^{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \]
Corollary 9.3.3.17 (Existence). Let $\kappa \leq \lambda $ be regular cardinals and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. Then there exists a diagram which exhibits $\widehat{U}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $U$.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{U} \ar [rr]^{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& } \]
Proof. Combine Theorem 8.7.4.1 with Remark 9.3.3.16. $\square$
In the situation of Corollary 9.3.3.17, the inner fibration $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ is uniquely determined up to equivalence. This is a consequence of the following universal property:
Corollary 9.3.3.18 (Uniqueness). Let $\kappa \leq \lambda $ be regular cardinals and suppose we are given a commutative diagram where $U$ and $\widehat{U}$ are inner fibrations and $H$ exhibits $\widehat{U}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $U$. Let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets which is $(\kappa ,\lambda )$-cocomplete, and let $\operatorname{Fun}'_{ / \operatorname{\mathcal{C}}}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ be the full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ spanned by those objects $F$ such that, for each vertex $C \in \operatorname{\mathcal{C}}$, the map of fibers $F_{C}: \widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \operatorname{\mathcal{D}}_{C}$ is $(\kappa ,\lambda )$-finitary. Then composition with $H$ induces an equivalence of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{U} \ar [rr]^{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \]
\[ \operatorname{Fun}'_{ / \operatorname{\mathcal{C}}}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}). \]
Proof. Combine Theorem 8.7.1.14 with Remark 9.3.3.16 (and Remark 9.2.9.8). $\square$
Corollary 9.3.3.19. Let $\kappa \leq \lambda $ be regular cardinals and suppose we are given a commutative diagram where $U$ and $\widehat{U}$ are inner fibrations and $H$ exhibits $\widehat{U}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $U$. If $U$ is a cocartesian fibration, then $\widehat{U}$ is also a cocartesian fibration. In this case, the diagram commutes up to isomorphism, where the vertical maps are the covariant transport representations of $U$ and $\widehat{U}$, respectively. Here $\mu > \lambda $ is any cardinal of exponential cofinality $\geq \lambda $ for which $U$ is essentially $\mu $-small.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{U} \ar [rr]^{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \]
\[ \xymatrix@C =50pt@R=50pt{ & \operatorname{\mathcal{C}}\ar [dl]_{\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}} \ar [dr]^{ \operatorname{Tr}_{\widehat{\operatorname{\mathcal{E}}}/\operatorname{\mathcal{C}}} } & \\ \operatorname{\mathcal{QC}}_{< \mu } \ar [rr]^{ \operatorname{Ind}_{\kappa }^{\lambda } } & & \operatorname{\mathcal{QC}}_{< \mu } } \]
Proof.
Combine Proposition with Remark 9.3.3.16.
$\square$
Corollary 9.3.3.20. Let $\kappa \leq \lambda $ be regular cardinals and suppose we are given a commutative diagram where $U$ and $\widehat{U}$ are inner fibrations and $H$ exhibits $\widehat{U}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $U$. If $U$ is a cocartesian fibration, then $\widehat{U}$ is also a cocartesian fibration. In this case, the diagram commutes up to isomorphism, where the vertical maps are the covariant transport representations of $U$ and $\widehat{U}$, respectively. Here $\mu > \lambda $ is any cardinal of exponential cofinality $\geq \lambda $ for which $U$ is essentially $\mu $-small.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{U} \ar [rr]^{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \]
\[ \xymatrix@C =50pt@R=50pt{ & \operatorname{\mathcal{C}}\ar [dl]_{\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}} \ar [dr]^{ \operatorname{Tr}_{\widehat{\operatorname{\mathcal{E}}}/\operatorname{\mathcal{C}}} } & \\ \operatorname{\mathcal{QC}}_{< \mu } \ar [rr]^{ \operatorname{Ind}_{\kappa }^{\lambda } } & & \operatorname{\mathcal{QC}}_{< \mu } } \]
Proof.
Combine Proposition with Remark 9.3.3.16.
$\square$
Remark 9.3.3.21. Let $\kappa \leq \lambda < \mu $ be regular cardinals, where $\mu $ has exponential cofinality $\geq \lambda $. Then the functor $\operatorname{Ind}_{\kappa }^{\lambda }: \operatorname{\mathcal{QC}}_{< \mu } \rightarrow \operatorname{\mathcal{QC}}_{< \mu }$ of Notation 9.3.3.8 can be characterized (up to isomorphism) by the conclusion of either Corollary 9.3.3.19 or Corollary 9.3.3.20.