$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.2.3.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $F$ is right cofinal.
- $(2)$
For every cocartesian fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$, the projection map $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}} \rightarrow \widetilde{\operatorname{\mathcal{D}}}$ is right cofinal.
- $(3)$
For every left fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$, the projection map $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}} \rightarrow \widetilde{\operatorname{\mathcal{D}}}$ is a weak homotopy equivalence.
- $(4)$
For every left fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$, if $\widetilde{\operatorname{\mathcal{D}}}$ is weakly contractible, then $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$ is weakly contractible.
Proof.
The implication $(1) \Rightarrow (2)$ follows from Proposition 7.2.3.12, the implication $(2) \Rightarrow (3)$ from Propositions 7.2.1.5 and 5.1.4.15, and the implication $(3) \Rightarrow (4)$ is immediate. We will complete the proof by showing that $(4)$ implies $(1)$. Assume that condition $(4)$ is satisfied; we wish to prove that $F$ is right cofinal. Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, where $\operatorname{\mathcal{E}}$ is an $\infty $-category. Note that $G$ is right cofinal, and therefore satisfies condition $(4)$ (Corollary 7.2.1.13). For each object $X \in \operatorname{\mathcal{E}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{X/}$ has an initial object and is therefore weakly contractible (Corollary 4.6.7.25). Since both $F$ and $G$ satisfy condition $(4)$, it follows that the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_{X/}$ is also weakly contractible. Allowing the object $X$ to vary, we see that $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is right cofinal (Theorem 7.2.3.1). Applying Proposition 7.2.1.21, we conclude that $F$ is also right cofinal.
$\square$