Theorem 9.2.4.3. Let $\kappa \leq \lambda $ be regular cardinals and suppose we are given a categorical pullback diagram of $\infty $-categories
9.6
\begin{equation} \begin{gathered}\label{equation:pullback-of-filtered-theorem} \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}
satisfying the following conditions:
- $(1)$
The $\infty $-categories $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}$, and $\operatorname{\mathcal{C}}_{+}$ are $\lambda $-filtered.
- $(2)$
The functors $F_{-}$ and $F_{+}$ are right cofinal.
- $(3)$
The $\infty $-categories $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $\kappa $-sequentially cocomplete, and the functors $F_{-}$ and $F_{+}$ preserve $\kappa $-sequential colimits (see Definition 9.2.3.1).
Then the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $\lambda $-filtered, and the functor $E_{-}$ and $E_{+}$ are right cofinal.
Proof of Theorem 9.2.4.3.
Let $\kappa \leq \lambda $ be regular cardinals and let
9.8
\begin{equation} \begin{gathered}\label{equation:pullback-of-kappa-filtered-infinity} \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}
be a categorical pullback diagram of $\infty $-categories which satisfies conditions $(1)$, $(2)$, and $(3)$ of Theorem 9.2.4.3, and let $E: \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}$ be the functor given by the composition $F_{+} \circ E_{+} = F_{-} \circ E_{-}$. For every $\lambda $-small diagram $q: K \rightarrow \operatorname{\mathcal{C}}_{\pm }$, the induced diagram of coslice $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ (\operatorname{\mathcal{C}}_{\pm })_{q/} \ar [r] \ar [d] & (\operatorname{\mathcal{C}}_{+})_{(E_{+} \circ q)/} \ar [d] \\ (\operatorname{\mathcal{C}}_{-})_{(E_{-} \circ q)/} \ar [r] & \operatorname{\mathcal{C}}_{(E \circ q)/} } \]
is also a categorical pullback square (Corollary 4.6.4.22) satisfying conditions $(1)$, $(2)$, and $(3)$ of Theorem 9.2.4.3: condition $(1)$ follows from Proposition 9.1.1.16, condition $(2)$ from Corollary 9.1.4.18, and condition $(3)$ from Corollary 7.1.7.5. Applying Lemma 9.2.4.5, we deduce that the $\infty $-category $( \operatorname{\mathcal{C}}_{\pm } )_{q/}$ is nonempty. Allowing $q$ to vary, we conclude that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $\lambda $-filtered.
We will complete the proof by showing that the functor $E_{-}$ is right cofinal (the assertion that $E_{+}$ is right cofinal follows by a similar argument). By virtue of Theorem 9.1.4.5, it will suffice to show that for every object $X \in \operatorname{\mathcal{C}}_{\pm }$, the induced map of coslice $\infty $-categories $(\operatorname{\mathcal{C}}_{\pm })_{X/} \rightarrow (\operatorname{\mathcal{C}}_{-})_{ E_{-}(X) / }$ is weakly right cofinal. Arguing as above, we can replace (9.8) by the diagram of coslice $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ (\operatorname{\mathcal{C}}_{\pm })_{X/} \ar [r] \ar [d] & (\operatorname{\mathcal{C}}_{+})_{E_{+}(X)/} \ar [d] \\ (\operatorname{\mathcal{C}}_{-})_{E_{-}(X)/} \ar [r] & \operatorname{\mathcal{C}}_{E(X)/} } \]
and thereby reduce to the problem of showing that the functor $E_{-}$ is weakly cofinal. Fix an object $Y \in \operatorname{\mathcal{C}}_{-}$; we wish to show that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm } \times _{ \operatorname{\mathcal{C}}_{-} } (\operatorname{\mathcal{C}}_{-})_{Y/}$ is nonempty. This follows by applying Lemma 9.2.4.5 to the outer rectangle in the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm } \times _{ \operatorname{\mathcal{C}}_{-} } (\operatorname{\mathcal{C}}_{-})_{Y/} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{\pm } \ar [r]^-{E_{+}} \ar [d]^{E_{-}} & \operatorname{\mathcal{C}}_{+} \ar [d]^{ F_{+} } \\ (\operatorname{\mathcal{C}}_{-})_{Y/} \ar [r] & \operatorname{\mathcal{C}}_{-} \ar [r]^-{ F_{-} } & \operatorname{\mathcal{C}}; } \]
note that the bottom horizontal composition is right cofinal by virtue of Corollary 9.1.4.16.
$\square$