Proposition 9.2.8.1. Let $\kappa \leq \lambda $ be regular cardinals, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories, and let $X$ be an object of $\operatorname{\mathcal{E}}$. Assume that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ are $(\kappa ,\lambda )$-cocomplete and that the functor $U$ is $(\kappa ,\lambda )$-finitary. If $U(X)$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$, then $X$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{E}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.2.8 Closure Properties of Compact Objects
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small filtered colimits. In this section, we collect some elementary results which can be used to establish the compactness of an object $C \in \operatorname{\mathcal{C}}$.
Proof. This is a special case of Lemma 8.4.8.7. $\square$
Corollary 9.2.8.2 (Compactness in Slice $\infty $-Categories). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be a $(\kappa ,\lambda )$-cocomplete $\infty $-category, and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. If $X$ is an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/F}$ whose image in $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact, then $X$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{C}}_{/F}$.
Proof. It follows from Corollary 7.1.4.22 that the $\infty $-category $\operatorname{\mathcal{C}}_{/F}$ admits $\lambda $-small $\kappa $-filtered colimits which are preserved by the right fibration $U: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}$, so the desired result is a special case of Proposition 9.2.8.1. $\square$
Under some additional assumptions, Corollary 9.2.8.2 has a counterpart for coslice $\infty $-categories (Corollary 9.2.8.11) which we will prove at the end of this section. The main ingredient we will need is the following:
Proposition 9.2.8.3 (Compactness in Oriented Fiber Products). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ be $\infty $-categories which are $(\kappa ,\lambda )$-cocomplete, and let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ be functors which are $(\kappa ,\lambda )$-finitary. Then:
The oriented fiber product $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is $(\kappa ,\lambda )$-cocomplete, and the projection maps $\operatorname{\mathcal{C}}_{-} \xleftarrow {\operatorname{ev}_{-}} \operatorname{\mathcal{C}}_{\pm } \xrightarrow {\operatorname{ev}_{+}} \operatorname{\mathcal{C}}_{+}$ are $(\kappa ,\lambda )$-finitary.
Let $C$ be an object of $\operatorname{\mathcal{C}}_{\pm }$, given by a triple $(C_{-}, C_{+}, u)$ where $C_{-}$ is an object of $\operatorname{\mathcal{C}}_{-}$, $C_{+}$ is an object of $\operatorname{\mathcal{C}}_{+}$, and $u: F_{-}(C_{-} ) \rightarrow F_{+}( C_{+} )$ is a morphism in $\operatorname{\mathcal{C}}$. If $C_{-}$, $C_{+}$, and $F_{-}(C_{-} )$ are $(\kappa ,\lambda )$-compact (as objects of $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$, respectively), then $C$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{\pm }$.
Remark 9.2.8.4. In the situation of Proposition 9.2.8.3, it is not necessary to assume that $F_{+}(C_{+} )$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$.
Proof of Proposition 9.2.8.3. Assertion $(1)$ follows from Corollary 7.1.9.5 (and does not require the assumption that $F_{+}$ is $(\kappa ,\lambda )$-finitary). We will prove $(2)$. Fix an uncountable cardinal $\mu $ of cofinality $\geq \lambda $ such that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ are locally $\mu $-small. We must show that the corepresentable functor $h^{C}: \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is $(\kappa ,\lambda )$-finitary. Let $h^{C_{-}}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$, $h^{C_{+}}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ and $h^{ F_{-}(C_{-} ) }: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ denote functors corepresented by $C_{-}$, $C_{+}$, and $F_{-}(C_{-})$, respectively. Let $\operatorname{ev}_{-}: \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ and $\operatorname{ev}_{+}: \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{+}$ be the projection maps. Applying Theorem 8.3.7.1 (and Remark 8.3.7.2), we obtain a (levelwise) pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ h^{C} \ar [r] \ar [d] & h^{C_{+}} \circ \operatorname{ev}_{+} \ar [d] \\ h^{C_{-}} \circ \operatorname{ev}_{-} \ar [r] & h^{ F_{-}(C_{-} )} \circ F_{+} \circ \operatorname{ev}_{+} } \]
in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}_{< \mu } )$. Since the collection of $(\kappa ,\lambda )$-finitary functors from $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{S}}_{< \mu }$ is closed under $\kappa $-small limits, it will suffice to show that the functors $h^{C_{+}} \circ \operatorname{ev}_{+}$, $h^{C_{-}} \circ \operatorname{ev}_{-}$, and $h^{ F_{-}(C_{-} )} \circ F_{+} \circ \operatorname{ev}_{+}$ are $(\kappa ,\lambda )$-finitary. Since the functors $\operatorname{ev}_{-}$, $\operatorname{ev}_{+}$, and $F_{+}$ are $(\kappa ,\lambda )$-finitary (Corollary 7.1.9.5), this follows from our assumption that the objects $C_{-}$, $C_{+}$, and $F_{-}( C_{-} )$ are $(\kappa ,\lambda )$-compact. $\square$
Corollary 9.2.8.5 (Compactness in Homotopy Fiber Products). Let $\kappa \leq \lambda $ be regular cardinals and suppose we are given a categorical pullback diagram of $\infty $-categories Assume that the $\infty $-categories $\operatorname{\mathcal{C}}_{+}$, $\operatorname{\mathcal{C}}_{-}$, and $\operatorname{\mathcal{C}}$ are $(\kappa ,\lambda )$-cocomplete and that the functors $F_{-}$ and $F_{+}$ are $(\kappa ,\lambda )$-finitary (so that $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-cocomplete and the functors $E_{+}$ and $E_{-}$ are $(\kappa ,\lambda )$-finitary; see Proposition 9.1.9.8). Let $C \in \operatorname{\mathcal{C}}_{\pm }$ be an object whose images in $\operatorname{\mathcal{C}}_{+}$, $\operatorname{\mathcal{C}}_{-}$, and $\operatorname{\mathcal{C}}$ are $(\kappa ,\lambda )$-compact. Then $C$ is $(\kappa ,\lambda )$-compact.
9.12
\begin{equation} \begin{gathered}\label{equation:compact-in-categorical-pullback} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm } \ar [r]^-{ E_{+} } \ar [d]^{ E_{-}} & \operatorname{\mathcal{C}}_{+} \ar [d]^{F_{+}} \\ \operatorname{\mathcal{C}}_{-} \ar [r]^-{F_{-}} & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
Proof. Our assumption that (9.12) is a categorical pullback square guarantees that $E_{-}$ and $E_{+}$ induce an equivalence of $\infty $-categories $E: \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$. It will therefore suffice to show that $E(C)$ is $(\kappa ,\lambda )$-compact when viewed as an object of the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$. In fact, $E(C)$ is even $(\kappa ,\lambda )$-compact when viewed as an object of the oriented fiber product $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$, by virtue of Proposition 9.2.8.3. $\square$
Proposition 9.2.8.6 (Compactness in Products). Let $\kappa \leq \lambda $ be regular cardinals, let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a collection of $(\kappa ,\lambda )$-cocomplete $\infty $-categories, and let $C = \{ C_ i \} _{i \in I}$ be an object of the product $\operatorname{\mathcal{C}}= \prod _{i \in I} \operatorname{\mathcal{C}}_{i}$. If $C$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$, then each $C_ i$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{i}$. The converse holds if the index set $I$ is $\kappa $-small.
Proof. Fix an uncountable cardinal $\mu $ such that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{i}$ is locally $\mu $-small. Then, for each index $i \in I$, we can choose a functor $\mathscr {F}_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ and a vertex $\eta _ i \in \mathscr {F}_ i( C_ i )$ which exhibits the functor $\mathscr {F}_ i$ as corepresented by the object $C_ i$ (see Definition 5.6.6.1). Enlarging $\mu $ if necessary, we may assume that it has cofinality $\geq \lambda $ and exponential cofinality larger than the cardinality of $I$, so that the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ admits $\lambda $-small colimits and $I$-indexed products (see Corollary 7.4.3.10, Variant 7.4.1.15 and Proposition 7.1.8.2). We can therefore choose a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ and a collection of natural transformations $u_ i: \mathscr {F} \rightarrow \mathscr {F}_ i |_{\operatorname{\mathcal{C}}}$ which exhibit $\mathscr {F}$ as a product of the collection $\{ \mathscr {F}_ i|_{\operatorname{\mathcal{C}}} \} _{i \in I}$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$. It follows that the natural transformations $u_ i$ induce a homotopy equivalence of Kan complexes $\mathscr {F}(C) \rightarrow \prod _{i \in I} \mathscr {F}_ i(C_ i)$ (see Example 7.6.1.16). We may therefore assume without loss of generality that the collection $\{ \eta _ i \} _{i \in I}$ is the image of some vertex $\eta \in \mathscr {F}(C)$. Using Remark 4.6.1.9, we see that $\eta $ exhibits the functor $\mathscr {F}$ as corepresented by $C$.
Assume first that $C$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$ and let $i$ be an element of $I$; we wish to show that $C_{i}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{i}$. Let $\delta : \operatorname{\mathcal{C}}_{i} \rightarrow \operatorname{\mathcal{C}}$ be the functor given by the identity map $\operatorname{id}: \operatorname{\mathcal{C}}_{i} \rightarrow \operatorname{\mathcal{C}}_ i$ together with the constant functors $\operatorname{\mathcal{C}}_ i \rightarrow \{ C_ j \} \hookrightarrow \operatorname{\mathcal{C}}_{j}$ for $j \neq i$. Our assumption that $C$ is $(\kappa ,\lambda )$-compact guarantees that the composite functor
\[ \operatorname{\mathcal{C}}_{i} \xrightarrow { \delta } \operatorname{\mathcal{C}}\xrightarrow { \mathscr {F} } \operatorname{\mathcal{S}}_{< \mu } \]
is $(\kappa ,\lambda )$-finitary. This composite functor can be identified with the product of $\mathscr {F}_{i}$ with the constant functor $\operatorname{\mathcal{C}}_{i} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ taking the value $X = \prod _{j \neq i} \mathscr {F}_ j(C_ j)$. Since the Kan complex $X$ is nonempty (it contains the vertex $\{ \eta _ j \} _{j \neq i}$), it follows that $\mathscr {F}_ i$ is a retract of $\delta \circ \mathscr {F}$ and is therefore also $(\kappa ,\lambda )$-finitary (Remark 9.2.2.13).
We now prove the converse. Assume that each $C_ i$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{C}}_{i}$. Then each of the functors $\mathscr {F}_ i$ is $(\kappa ,\lambda )$-finitary, and therefore restricts to a $(\kappa ,\lambda )$-finitary functor $\mathscr {F}_{i}|_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ (Proposition 9.1.9.8). If $I$ is $\kappa $-small, then the formation of $I$-indexed products commutes with $\lambda $-small $\kappa $-filtered colimits in the $\infty $-category $\operatorname{\mathcal{S}}_{< \mu }$ (Theorem 9.1.5.9), so the functor $\mathscr {F} = \prod _{i \in I} \mathscr {F}_ i|_{\operatorname{\mathcal{C}}}$ is also $(\kappa ,\lambda )$-finitary: that is, the object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact. $\square$
Corollary 9.2.8.7 (Compactness in Sequential Limits). Let $\kappa \leq \lambda $ be uncountable regular cardinals and suppose we are given a tower of $\infty $-categories Assume that:
Each of the $\infty $-categories $\operatorname{\mathcal{C}}_{n}$ is $(\kappa ,\lambda )$-cocomplete.
Each of the functors $U_{n}: \operatorname{\mathcal{C}}_{n+1} \rightarrow \operatorname{\mathcal{C}}_ n$ is $(\kappa ,\lambda )$-finitary.
Each of the functors $U_ n$ is an isofibration, so that the inverse limit $\varprojlim (\operatorname{\mathcal{C}}_{n})$.
\[ \cdots \rightarrow \operatorname{\mathcal{C}}_{3} \xrightarrow { U_2} \operatorname{\mathcal{C}}_2 \xrightarrow { U_1 } \operatorname{\mathcal{C}}_1 \xrightarrow {U_0} \operatorname{\mathcal{C}}_0. \]
Let $C$ be an object of the inverse limit $\varprojlim \operatorname{\mathcal{C}}_{n}$, given by a sequence of objects $\{ C_ n \in \operatorname{\mathcal{C}}_ n \} $ satisfying $F_{n}(C_{n+1} ) = C_{n}$. If each $C_{n}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{n}$, then $C$ is $(\kappa ,\lambda )$-compact object of $\varprojlim (\operatorname{\mathcal{C}}_{n})$.
Proof. Set $\operatorname{\mathcal{C}}= \prod _{n} \operatorname{\mathcal{C}}_{n}$, and let $S: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be the shift functor (given on objects by the construction $\{ D_ n \} _{n \geq 0} \mapsto \{ U_{n}( D_{n+1} ) \} _{n \geq 0}$). By virtue of Corollary 4.5.6.19, the diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \varprojlim \operatorname{\mathcal{C}}_{n} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ (\operatorname{id}, \operatorname{id}) } \\ \operatorname{\mathcal{C}}\ar [r]^-{ (\operatorname{id},S) } & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}} \]
is a categorical pullback square, in which the maps are given by $(\kappa ,\lambda )$-finitary functors. By virtue of Corollary 9.2.8.5, to show that the sequence $C = \{ C_ n \} _{n \geq 0}$ is $(\kappa ,\lambda )$-compact as an object of the inverse limit $\varprojlim (\operatorname{\mathcal{C}}_{n})$, it will suffice to show that it is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{C}}$ and that $(C,C)$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}$. Both assertions follow from Proposition 9.2.8.6 (since $\kappa $ is uncountable and each $C_ n$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{n}$). $\square$
Warning 9.2.8.8. In the statement of Corollary 9.2.8.7, the assumption that $\kappa $ is uncountable cannot be omitted.
Proposition 9.2.8.9 (Compactness in Diagram $\infty $-Categories). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete, and let $K$ be a simplicial set. Then:
The diagram $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is $(\kappa ,\lambda )$-cocomplete.
Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. If $K$ is $\kappa $-small and $F(k) \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact for each vertex $k \in K$, then $F$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$.
Proof. Assertion $(1)$ follows from Proposition 7.1.8.2. We will prove $(2)$. Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram having the property that, for each vertex $k \in K$, the object $F(k) \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact. If $K'$ is a simplicial set with a map $K' \rightarrow K$, we will say that $K'$ is good if $F|_{K'}$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{Fun}(K', \operatorname{\mathcal{C}})$. To complete the proof, it will suffice to show that if $K'$ is $\kappa $-small, then it is good.
Note that $\operatorname{Fun}(K',\operatorname{\mathcal{C}})$ can be realized as the inverse limit of a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{Fun}( \operatorname{sk}_{2}(K'), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{sk}_{1}(K'), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{sk}_{0}(K'), \operatorname{\mathcal{C}}), \]
where the transition maps are isofibrations (Corollary 4.4.5.3). Each of the $\infty $-categories appearing in this tower is $(\kappa ,\lambda )$-cocomplete and each of the transition functors is $(\kappa ,\lambda )$-finitary. It will therefore suffice to show that each of the skeleta $\operatorname{sk}_{n}(K')$ is good: this is immediate if $\kappa = \aleph _0$ (the assumption that $K'$ is finite guarantees that it coincides with $\operatorname{sk}_{n}(K')$ for $n \gg 0$), and follows from Corollary 9.2.8.7 when $\kappa > \aleph _0$. We may therefore replace $K'$ by $\operatorname{sk}_{n}(K')$ and thereby reduce to the case where the simplicial set $K$ has dimension $\leq n$, for some integer $n \geq -1$.
We now proceed by induction on $n$. If $n = -1$, then the simplicial set $K'$ is empty and the result is clear. Otherwise, let $K'' = \operatorname{sk}_{n-1}(K')$ be the $(n-1)$-skeleton of $K$ and let $S$ be the collection of all nondegenerate $n$-simplices of $K'$ (which is $\kappa $-small, by virtue of our assumption that $K'$ is $\kappa $-small). Applying Proposition 1.1.4.12, we obtain a pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( K', \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}( K'', \operatorname{\mathcal{C}}) \ar [d] \\ \prod _{\sigma \in S} \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \ar [r] & \prod _{\sigma \in S} \operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}}). } \]
Since the horizontal maps in this diagram are isofibrations (Corollary 4.4.5.3), it is also a categorical pullback square (Corollary 4.5.2.28). Moreover, each of the $\infty $-categories in the diagram is $(\kappa ,\lambda )$-cocomplete of the transition functors is $(\kappa ,\lambda )$-finitary. Combining our inductive hypothesis, Corollary 9.2.8.5, and Proposition 9.2.8.6, we can reduce to the case where $K' = \Delta ^ n$ is a standard simplex.
If $n = 0$, the desired result follows from our assumption that $F$ carries vertices of $K$ to $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$. If $n=1$, then we can identify $\operatorname{Fun}(K', \operatorname{\mathcal{C}})$ with the oriented fiber product $\operatorname{\mathcal{C}}\vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, and the desired result follows from Proposition 9.2.8.3. We may therefore assume that $n \geq 2$. Let $\operatorname{Spine}[n] \subseteq \Delta ^ n$ be as in Example 1.5.7.7. The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne and therefore induces a trivial Kan fibration $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{Spine}[n], \operatorname{\mathcal{C}})$. The desired result now follows from our inductive hypothesis, since the simplicial set $\operatorname{Spine}[n]$ has dimension $1$. $\square$
Warning 9.2.8.10. The converse of Proposition 9.2.8.9 is false in general. For example, let $K = \Delta ^1 / \operatorname{\partial \Delta }^1$ be the simplicial circle (Example 1.5.7.11) and let $\operatorname{\mathcal{C}}$ be (the nerve of) the category $\operatorname{ Ab }$ of abelian groups. Then $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ can be identified with (the nerve of) the category of pairs $(M,u)$, where $M$ is an abelian group and $u: M \rightarrow M$ is an endomorphism of $M$. In this case, $(M,u)$ is a compact object of $\operatorname{\mathcal{C}}$ if and only if $M$ is finitely generated as a module over the polynomial ring $\operatorname{\mathbf{Z}}[u]$. (see Proposition 9.2.0.1). This condition does not guarantee that $M$ is finitely generated as an abelian group (for example, the polynomial ring $\operatorname{\mathbf{Z}}[u]$ itself is not finitely generated as an abelian group).
Corollary 9.2.8.11 (Compactness in Coslice $\infty $-Categories). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be a $(\kappa ,\lambda )$-cocomplete $\infty $-category, and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a $\kappa $-small diagram which carries each vertex of $K$ to a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$. If $X$ is an object of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{F/}$ whose image in $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact, then $X$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{C}}_{F/}$.
Proof. Theorem 4.6.4.19 supplies an equivalence of $\infty $-categories
\[ \operatorname{\mathcal{C}}_{F/} \hookrightarrow \{ F\} \vec{\times }_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}. \]
The desired results now follow by combining Propositions 9.2.8.9 and 9.2.8.3. $\square$