Theorem 9.1.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are filtered. Then $F$ is right cofinal if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the induced functor $F_{C/}: \operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{D}}_{ F(C) / }$ is weakly right cofinal.
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Proof of Theorem 9.1.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between filtered $\infty $-categories. Assume that, for every object $C \in \operatorname{\mathcal{C}}$, the functor $F_{C/}: \operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{D}}_{ F(C) / }$ is weakly right cofinal; we wish to show that $F$ is right cofinal (the converse follows Corollary 9.1.4.19 and Example 9.1.4.2). By virtue of Theorem 7.2.3.1, it will suffice to show that for each object $X \in \operatorname{\mathcal{D}}$, the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ X/ }$ is weakly contractible. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be given by projection onto the first factor. We will complete the proof by showing that the left fibration $U$ satisfies the criterion of Proposition 9.1.3.8, so that $\widetilde{\operatorname{\mathcal{C}}}$ is filtered. Fix an object $C \in \operatorname{\mathcal{C}}$ and a finite diagram $f: K \rightarrow \{ C\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}}$. We wish to show that there is a morphism $w: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$ such that the composite map
\[ K \xrightarrow {f} \{ C\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}} \xrightarrow { w_{!} } \{ C'\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}} \]
is nullhomotopic, where $v_{!}$ is the covariant transport functor associated to the left fibration $U$. Note that, since the $\infty $-category $\operatorname{\mathcal{D}}_{X/}$ is filtered (Proposition 9.1.1.17), the analogous condition is satisfied for the left fibration $V: \operatorname{\mathcal{D}}_{X/} \rightarrow \operatorname{\mathcal{D}}$; that is, we can choose a morphism $u: F(C) \rightarrow D$ in $\operatorname{\mathcal{D}}$ for which the composite map
\[ K \xrightarrow {f} \{ F(C) \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/} \xrightarrow { u_{!} } \{ D \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}. \]
We now complete the proof by taking $w: C \rightarrow C'$ to be any morphism of $\operatorname{\mathcal{C}}$ having the property that $F(w)$ factors through $u$ (the existence of which is guaranteed by our assumption that $F_{C/}$ is weakly right cofinal; see Remark 9.1.4.6. $\square$