Proof.
Suppose first that $\operatorname{\mathcal{C}}$ is filtered. Assertion $(a)$ follows from Proposition 7.2.4.9. To prove $(b)$, suppose we are given left fibrations $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$, $V_0: \widetilde{\operatorname{\mathcal{C}}}_{0} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$, and $V_{1}: \widetilde{\operatorname{\mathcal{C}}}_{1} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$, where $\widetilde{\operatorname{\mathcal{C}}}$, $\widetilde{\operatorname{\mathcal{C}}}_{0}$, and $\widetilde{\operatorname{\mathcal{C}}}_{1}$ are weakly contractible. Applying Theorem 7.2.6.1, we deduce that the $\infty $-categories $\widetilde{\operatorname{\mathcal{C}}}$, $\widetilde{\operatorname{\mathcal{C}}}_{0}$, and $\widetilde{\operatorname{\mathcal{C}}}_{1}$ are filtered. Applying Corollary 7.2.6.2 to the diagram of left fibrations
\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}}_{0} \times _{ \widetilde{\operatorname{\mathcal{C}}} } \widetilde{\operatorname{\mathcal{C}}}_{1} \ar [r] \ar [d] & \widetilde{\operatorname{\mathcal{C}}}_{0} \ar [d]^{V_0} \\ \widetilde{\operatorname{\mathcal{C}}}_{1} \ar [r]^-{V_1} & \widetilde{\operatorname{\mathcal{C}}}, } \]
we conclude that the fiber product $\widetilde{\operatorname{\mathcal{C}}}_{0} \times _{ \widetilde{\operatorname{\mathcal{C}}}} \widetilde{\operatorname{\mathcal{C}}}_{1}$ is also filtered; in particular, it is weakly contractible (Proposition 7.2.4.9).
We now prove the converse. Assume that $\operatorname{\mathcal{C}}$ satisfies conditions $(a)$ and $(b)$; we wish to show that $\operatorname{\mathcal{C}}$ is filtered. We will prove this using the criterion of Lemma 7.2.5.13. Fix an integer $n \geq 0$ and a diagram $e: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the coslice $\infty $-category $\operatorname{\mathcal{C}}_{e/}$ is nonempty. In fact, we will prove the following stronger assertion: for every simplicial subset $K \subseteq \operatorname{\partial \Delta }^{n}$, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{ e_ K / }$ is weakly contractible, where $e_{K}$ denotes the restriction $e|_{K}$. Our proof proceeds by induction on the number of nondegenerate simplices of $K$. If $K = \emptyset $, then the desired result follows from assumption $(a)$. If $K$ is not isomorphic to a standard simplex, then we can use Proposition 1.1.4.12 to write $K$ as a union $K(0) \cup K(1)$, where $K(0), K(1) \subsetneq K$ are proper simplicial subsets. Setting $K(01) = K(0) \cap K(1)$, we have a pullback diagram of left fibrations
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{ e_ K / } \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{ e_{K(0)} / } \ar [d] \\ \operatorname{\mathcal{C}}_{ e_{ K(1) } / } \ar [r] & \operatorname{\mathcal{C}}_{ e_{ K(01) } / }, } \]
where the $\infty $-categories $\operatorname{\mathcal{C}}_{ e_{K(0)} / }$, $\operatorname{\mathcal{C}}_{ e_{K(1)} / }$, and $\operatorname{\mathcal{C}}_{ e_{K(01)} / }$ are weakly contractible by virtue of our inductive hypothesis. Applying $(b)$, we deduce that $\operatorname{\mathcal{C}}_{ e_ K / }$ is weakly contractible. We may therefore assume without loss of generality that $K \simeq \Delta ^{m}$ is a standard simplex. In particular, $K$ contains a final vertex $v$ for which the inclusion $\{ v\} \hookrightarrow K$ is right anodyne (Example 4.3.7.11), so that the restriction map $\operatorname{\mathcal{C}}_{ e_ K / } \rightarrow \operatorname{\mathcal{C}}_{ e(v) / }$ is a trivial Kan fibration (Corollary 4.3.6.13). It will therefore suffice to show that the $\infty $-category $\operatorname{\mathcal{C}}_{ e(v) / }$ is weakly contractible. This follows from Corollary 4.6.7.25, since the $\infty $-category $\operatorname{\mathcal{C}}_{e(v) / }$ has an initial object (Proposition 4.6.7.22).
$\square$