Proposition 9.1.3.8. Let $\kappa $ be an infinite cardinal, let $\operatorname{\mathcal{C}}$ be a $\kappa $-filtered $\infty $-category, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{E}}$ is $\kappa $-filtered.
- $(2)$
For every object $C \in \operatorname{\mathcal{C}}$ and every diagram $e: K \rightarrow \operatorname{\mathcal{E}}_{C}$, where $K$ is $\kappa $-small, there exists an object $Y \in \operatorname{\mathcal{E}}$ and a natural transformation from $e$ to the constant diagram $\underline{Y}$ (in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{E}})$).
- $(3)$
For every object $C \in \operatorname{\mathcal{C}}$ and every diagram $e: K \rightarrow \operatorname{\mathcal{E}}_{C}$, where $K$ is $\kappa $-small, there exists a morphism $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$ satisfying the following condition:
- $(\ast )$
There is an object $Y \in \operatorname{\mathcal{E}}_{D}$ and a natural transformation $\alpha : e \rightarrow \underline{Y}$ (in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{E}})$) such that $U(\alpha )$ is the constant natural transformation $\underline{f}: \underline{C} \rightarrow \underline{D}$ (in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$).
Proof of Proposition 9.1.3.8.
The implications $(1) \Rightarrow (2)$ and $(3) \Rightarrow (2)$ are immediate from the definitions, and the implication $(2) \Rightarrow (1)$ follows from Lemma 9.1.3.1. We will complete the proof by showing that $(2) \Rightarrow (3)$. Let $C \in \operatorname{\mathcal{C}}$ be an object and let $e: K \rightarrow \operatorname{\mathcal{E}}_{C}$ be a diagram, where $K$ is a $\kappa $-small simplicial set. It follows from $(2)$ that there exists an object $X \in \operatorname{\mathcal{E}}$ and a natural transformation $\beta : e \rightarrow \underline{X}$ in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{E}})$. Set $C' = U(X)$ and form a pushout diagram of simplicial sets
\[ \xymatrix { (\{ 0\} \times K) \coprod ( \{ 1\} \times K ) \ar [r] \ar [d] & \{ C\} \coprod \{ C' \} \ar [d] \\ \Delta ^1 \times K \ar [r]^{q} & L. } \]
By construction, $U(\beta )$ corresponds to a diagram in $\operatorname{\mathcal{C}}$ which factors as a composition
\[ \Delta ^1 \times K \xrightarrow {q} L \xrightarrow { s } \operatorname{\mathcal{C}}. \]
Since $L$ is $\kappa $-small and $\operatorname{\mathcal{C}}$ is $\kappa $-filtered, the diagram $s$ admits an extension $\overline{s}: L^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Let $D \in \operatorname{\mathcal{C}}$ be the image under $\overline{s}$ of the cone point of $L^{\triangleright }$. Restricting $\overline{s}$ to the edges $\{ C\} ^{\triangleright } \subseteq L^{\triangleright }$ and $\{ C'\} ^{\triangleright } \subseteq L^{\triangleright }$, we obtain a pair of morphisms $f: C \rightarrow D$ and $g: C' \rightarrow D$. Note that $q$ extends to a morphism $\overline{q}: \Delta ^2 \times K \rightarrow L^{\triangleright }$, carrying $\{ 2\} \times K$ to the cone point of $L^{\triangleright }$. Let us identify the composition $\overline{s} \circ \overline{q}$ with a $2$-simplex $\sigma $ of the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$, which we depict as a diagram
\[ \xymatrix { & \underline{C}' \ar [dr]^{ \underline{g} } & \\ \underline{C} \ar [ur] \ar [rr]^{ \underline{f} } & & \underline{D}. } \]
Since $U$ is a cocartesian fibration, we can write $g = U( \widetilde{g} )$ for some morphism $\widetilde{g}: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$. Let $\gamma : \underline{X} \rightarrow \underline{Y}$ denote the image of $\widetilde{g}$ under the diagonal map $\operatorname{\mathcal{E}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{E}})$. Using Corollary 4.1.4.2, we can lift $\sigma $ to a $2$-simplex
\[ \xymatrix { & \underline{X} \ar [dr]^{ \gamma } & \\ e \ar [ur]^{ \beta } \ar [rr]^{\alpha } & & \underline{Y} } \]
of the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{E}})$. By construction, we have $U( \alpha ) = \underline{f}$, so the morphism $f$ satisfies condition $(\ast )$.
$\square$