Remark 9.4.1.14. In the situation of Definition 9.4.1.12, suppose that $\operatorname{\mathcal{C}}$ is an $\infty $-category. Then $H$ is fully faithful if and only if, for every morphism $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the induced map
\[ \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^1 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}} \]
is fully faithful (see Variant 4.8.6.19). In particular, if $H$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$, then $H$ is fully faithful.