$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 9.1.5.2. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category and suppose that we are given a collection of cocartesian fibrations $\{ U_ i: \operatorname{\mathcal{E}}_ i \rightarrow \operatorname{\mathcal{C}}\} _{i \in I}$ indexed by a set $I$. Assume that each of the $\infty $-categories $\operatorname{\mathcal{E}}_{i}$ is filtered and that $\operatorname{\mathcal{C}}$ satisfies the following condition:
- $(\ast _{I})$
For every object $C \in \operatorname{\mathcal{C}}$ and every collection of morphisms $\{ f_ i: C \rightarrow D_ i \} _{i \in I}$, there exists a morphism $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$ which factors through each $f_ i$.
Then the pullback $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(I, \operatorname{\mathcal{C}}) } (\prod _{i \in I} \operatorname{\mathcal{E}}_{i})$ is filtered.
Proof of Lemma 9.1.5.2.
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ denote the projection onto the first factor. It follows from Remark 5.1.4.7 that $U$ is a cocartesian fibration. We will show that $U$ satisfies condition $(3)$ of Proposition 9.1.3.8. Suppose we are given an object $C \in \operatorname{\mathcal{C}}$ and a diagram $e: K \rightarrow \operatorname{\mathcal{E}}_{C}$, where $K$ is a finite simplicial set. For each $i \in I$, our assumption that $\operatorname{\mathcal{E}}_ i$ is $\kappa $-filtered guarantees that we can choose a morphism $f_ i: C \rightarrow D_ i$ of $\operatorname{\mathcal{C}}$ for which the the composite diagram
\[ K \xrightarrow {e} \{ C\} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \{ C\} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}_ i \xrightarrow { f_{i!} } \{ D_ i \} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}_ i \]
admits an extension to $K^{\triangleright }$. Using condition $(\ast _{I})$, we can choose a morphism $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$ which factors through each of the morphisms $f_{i}$. Then the diagram
\[ K \xrightarrow {e} \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\xrightarrow { f_{!} } \{ D \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}} \]
admits an extension to $K^{\triangleright }$.
$\square$