Proposition 9.1.4.17. Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be filtered $\infty $-categories. Suppose we are given functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, where $G$ is a left fibration. Then $F$ is right cofinal if and only if $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is right cofinal.
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Proof. Assume that $G \circ F$ is right cofinal; we will prove that $F$ is right cofinal (the converse follows from Proposition 7.2.1.6, since $G$ is right cofinal by virtue of Corollary 9.1.4.14). Note that $F$ factors as a composition
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\xrightarrow {V} \operatorname{\mathcal{D}}, \]
where $F'$ is given given by the pair of functors $(\operatorname{id}_{\operatorname{\mathcal{C}}}, F)$ and $V$ is given by projection onto the second factor. Here $V$ is the pullback of the right cofinal functor $(G \circ F)$ along the left fibration $G$, and is therefore right cofinal by virtue of Proposition 7.2.3.10. It will therefore suffice to show that $F'$ is right cofinal (Proposition 7.2.1.6).
Note that $V$ is a weak homotopy equivalence (Proposition 7.2.1.5). Since $\operatorname{\mathcal{D}}$ is weakly contractible (Proposition 9.1.1.18), it follows that $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also weakly contractible. Let $G': \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be the projection map onto the second factor. Then $G'$ is a pullback of $G$, and is therefore a left fibration. Applying Corollary 9.1.4.14, we conclude that $G'$ is right cofinal. Since the composition $G' \circ F' = \operatorname{id}_{\operatorname{\mathcal{C}}}$ is also right cofinal, Proposition 7.2.1.6 guarantees that $F'$ is right cofinal as desired. $\square$