Example 9.1.4.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
The functor $F$ is weakly right cofinal if and only if, for each $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is nonempty.
The functor $F$ is right cofinal if and only if, for each $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is weakly contractible (Theorem 7.2.3.1).
In particular, if $F$ is right cofinal, then it is weakly right cofinal.