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Remark 7.2.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. For every object $X \in \operatorname{\mathcal{D}}$, the slice and coslice diagonal morphisms of Construction 4.6.4.13 induce categorical equivalences

\[ \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \quad \quad \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/} \hookrightarrow \{ X\} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{C}} \]

(Example 5.1.6.7). We can therefore reformulate Theorem 7.2.3.1 as follows:

$(1')$

The morphism $F$ is left cofinal if and only if, for every object $X \in \operatorname{\mathcal{D}}$, the simplicial set $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} $ is weakly contractible.

$(2')$

The morphism $F$ is right cofinal if and only if, for every object $X \in \operatorname{\mathcal{D}}$, the simplicial set $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ is weakly contractible.