Remark 4.8.5.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is full (in the sense of Definition 4.8.5.1) if and only if the induced functor of homotopy categories $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is full (in the usual category-theoretic sense).
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